1(a)

Clarity – 2

Style – 2

Comment – this is how I think of the proof upon seeing the question and pondering it for a second or two. Provided you’ve previously defined an open ball, this is great.

1(b)

Clarity – 1

Style – 1

Having extra definitions within proofs almost tautologically makes things more difficult to understand.

1(c)

Clarity – 0

Style – 1

This feels like reading C code as opposed to Python code 🙂 All those d’s and eta’s for me make it much harder to get a feel for the idea behind proof.

2(a)

Clarity – 0

Style – 1

I would make very similar comments to 1(c). Because I couldn’t quite remember how the proof went, I skipped this after a few seconds and read the ones below 🙂

2(b)

Clarity – 1

Style – 2

Still too much LaTeX 🙂 But this is definitely a lot easier than 2(a), especially given the first two sentences.

2(c)

Clarity -2

Style -2

This is the proof I went to to jog my memory, and it was very clear. The visual image stirred up by the use of open balls is to me very compelling.

3(a)

Clarity – 2

Style -2

Again, I had to stop and find the argument for myself, which probably took 20 seconds. I then read this and it felt like I was reading my own thoughts. Very clear.

3(b)

Clarity – 1

Style -1

I felt this needlessly repeated itself. Also, you said (a_n) converges without saying that it must converge somewhere in X as you did in (a). This small statement, putting bounds on where this mysterious point of convergence could be, adds a huge amount of concreteness. After all, that’s all we care about – where the point of convergence is.

3(c)

Clarity – 1

Style -1

Rudin has a lot of these A implies B, B implies C, C implies D, etc. proofs, and personally I feel it takes away the clarity. Here we have 3 sentences which all begin with the word ‘Since’ and two with the word ‘So’, which for me horribly disrupts the flow and overall picture of the argument.

4(a)

Clarity – 2

Style – 2

Very easily jogged my memory. The first paragraph is excellent.

4(b)

Clarity – 1

Style – 2

I didn’t mind this proof, it is clear enough. Making the substitution for z and y brings some concreteness.

4(c)

Clarity –0

Style – 1

My least favourite 🙂 Why do you need to bring in \theta ?

Meta-comment: where are the open balls in Q4? 😀

5(a)

Clarity – 2

Style – 2

It took me a minute to figure why we needed injectivity. This proof highlights this straight away and gets straight to the point.

5(b)

Clarity – 1

Style – 1

Not too bad, but similar problems to 3(c): while 5(a) has only 3 occurrences of ‘so’, this proof has 4, plus one ‘since’ and one ‘then’ :D.

5(c)

Clarity – 0

Style – 1

I said for 4(b) I liked the definition of z and y. Perhaps this is because they’re the 2 points crucial to the proof and are worth focusing the mind’s eye on. Here though, defining y and z is not necessary and just puts extra strain on the memory.

Overall Comments: I think it’s fair to say I enjoyed the proofs which:

Appealed to geometrical or ‘higher-order’ ideas. I again make the link to computer programming: while C is closer to the ‘machinations’ of the hardware, higher-order constructs like classes in Python make everything so much simpler to quickly understand 🙂

Length: In almost every case, the shortest proof with the least LaTeX was my favourite.

]]>Problem 1.

a) Clarity: 1

Style: 1

b) Clarity: 0

Style: -1

c)Clarity: -2

Style: -2

Comments: The first proof is the easiest to read by far. The notation a,b for A and B resp. is good. Introducing the definitions as in 1(b) is awkward and cumbersome. The writing style of 1(c) is strange (“therefore” in the second sentence is unclear). The last one is reminiscent of students struggling with the definitions.

Problem 2.

a) Clarity: -1

Style: -1

b) Clarity: 1

Style: 1

c)Clarity: 1

Style: 1

Comments: 2(b) is good in that it states what we are trying to prove in the notation _to be used_! This is always helpful. 1(a) is a mess, but I feel that the lack of line breaks is not helping

Problem 3.

a) Clarity: 2

Style: 1

b) Clarity: 0

Style: 1

c)Clarity: 2

Style: 2

Comments: 3(a) and 3(c) are good. I prefer (c) in that the letter _a_ is used but I prefer (a) in that the limit notation is clearer!

Problem 4.

a) Clarity: 0

Style: -2

b) Clarity: 1

Style: 1

c)Clarity: 2

Style: 2

Comments: The first one contains the implication symbol, which is too many symbols and overall I had a hard time reading it. (b) Is better but too many primes. The third is the best as it uses the fewest symbols and variables. One of the worst types of mistakes in mathematical writing is to introduce new symbols when it is unnecessary.

Problem 5.

a) Clarity: 2

Style: 2

b) Clarity: 0

Style: 1

c)Clarity: -1

Style: -2

Comments: The first is the best as it has the fewest words. The second takes too long to state f(a) = f(b). 5(c) is the worst as it doesn’t put the f(y) = f(z)! Also it states the definition of intersection which is too verbose.

]]>Thank you so much for thinking about these issues seriously.

1a. Clarity 1, style -1.

Notation is screwed up (confusion between radius and centre of the ball), and the radiuses are written a and b, a notation which suggests elements of the sets A and B, which only adds to the confusion.

1b. Clarity 2, style 2

(what we are doing could perhaps be stated at the start and not at the end ? I don’t know. In any case it’s a clean proof.).

1c. Clarity 0, style -2

Too many useless notations : clutters working memory. Unnecessarily complex.

2a. Clarity -1, style -2.

Inverse images tend to be confusing : one should be very consistent with the notations (using x, x’,… for elements of X and y, y’,… for elements of Y) and here they are very bad. using y for an element of X should be forbidden and using u is not very helpful.

2b. Clarity 0, style -1.

Confusion between centre and radius again, but here could easily pass unnoticed (no confusing notation). However one should use ball notation everywhere and not just occasionally : the constant switch between balls and inequalities is painful.

2c. Clarity 2, style 2.

Here I am not disturbed by the final recapitulation. I think stating the goal at the start is not important here (as opposed to 1b) as the proposition is less elementary (students would be more familiar with openness), but a final recapitulation is clean and helpful.

3a. Clarity 2, style 2.

(But the sentence « suppose lim x_n = x » is superfluous : one could write « .. has a limit x in X » in the previous sentence.)

3b. Clarity 0, style -2.

Writing a_n and a kind of presupposes that both sequence and limit are in A : confusing. In this context defining the limit as a wthout stating which set it a priori belongs to is downright perverse. (I dislike sentences like « Since P, P’ and P”, we have Q » : in a quick reading I tend to see P’ as a consequence and need to start again).

3c. Clarity 1, style -1.

Again, writing a_n and a kind of presupposes that both sequence and limit are in A : confusing. But less serious since a is defined as merely being in X and the argument is clearer. The expressions « tends to a limit in A » and especially « càontains all its limit points » are sloppy.

4a. Clarity 1, style -1.

4b. Clarity 1, style -1.

There is a typo f(y’) instead of f(y). The use of the notation y in the first step is helpful, but the use of the notation z not that much and only hides the typo ; and the notation y is unnecessary in the second step.

4c. Clarity 2, style 2.

The use of y for an element of X does not disturb me here although I would probably have written x’. The proof is straightforward and keeps notations at a minimum.

5a. Clarity 1, style -1.

The use of x, y and z causes cognitive dissonance.

5b. Clarity 2, style 2.

5c. Clarity 1, style -2.

Cognitive dissonance and too many notations. The insistence on the meaning of intersection is superfluous : it breaks the flow of the argument. Plus, I dislike sentences like « Since P, P’ and P”, we have Q », cf 3b.

Prob1

1(b) and 1(c) look lengthy and wordy. Besides, the “open ball” language is easier to read than the argument.

Rating Clarity Style

1(a) 2 2

1(b) 1 1

1(c) 0 0

Prob2

The “open ball” language in 2(b) and 2(c) is much more straightforward. For the sake of proving “open set”, one does finally need “open balls”. I need to “translate” as for understanding the proof.

2(a) 1 1

2(b) 2 2

2(c) 2 2

Prob3

The third one is more of "algorithm" style. The second one is wordy.

3(a) 1 1

3(b) 0 0

3(c) 2 2

Prob4

4(a) and 4(b) are ugly long and hard to read. Every step in 4(c) is straigtforward and immediately from the definition.

4(a) -1 -2

4(b) -1 -2

4(c) 2 2

Prob5

5(c) is wordy. 5(a) is straightforward to read.

5(a) 2 2

5(b) 1 1

5(c) 0 0

All of the proofs were straight forward but I found the wording of some distracting from the core idea.

Problem 3 was the most confusing because I wasn’t that familiar with the definition of a complete metric space and the fact that a closed set contains all its limit points.

1. 2, -1, -2

2. -2, 2, 2

3. -2, -2, 2

4. -2, 2, -1

5. 2, 2, -1

Ratings given as [clarity, style]

I don’t know if it makes a difference but problems 1, 3 and 4 were obvious to me (and the proofs went along the lines that I expected), whereas 2 and 5 were non-obvious.

1a: [2,2]

1b: [1, -1] choice of terminology was poor – why r prime?

1c: [0, 0] rather than leveraging the concept of an open ball this one was more from first principles, but this meant too many etas, thetas, deltas and nus were going on which I was struggling to hold in my head

2a: [0, 0] introduced theta for no good reason I could see.

2b: [0, 1] flipped between using open balls and a more direct definition of openness – epsilon got too much prominence in the proof, slightly clouding things for me. The “i.e.” section added nothing and only served to distract me slightly.

2c: [1, 1]

3a: [2, 2] didn’t like the “suppose” when what you’re doing is giving something a name, I would have used “call”

3b: [2, 1] again, the “that is” section seemed unecessary and just served to make me check whether I agreed with the “that is” rather than moving on with the proof. I would delete the words “we have than a_n converges”. Also, by this point I’m started to get irrationally irritated by “we are done” at the end of proofs.

3c: [2, 2]

4a: [2, 2]

4b: [-1, -2] the choice of names for things is all over the place. Why let y = f(x) – why not just use f(x) throughout? The substitution introduces an extra name to keep in my mind for only the tiniest increase in brevity of some of the expressions. Givening things names is only useful when the name is significant shorter than the thing it refers to, or some kind of summary of a number of steps.

4c: [2, 2] theta is mildly superfluous.

5a: [1, 2] I can see all of the steps work, but this doesn;t feel obvious from this proof.

5b: [2, 2] somehow this one makes me feel the results is right. Maybe it’s just that now I have two proofs? Or maybe it’s the use of a and b rather than y and z has let me keep track of things in my head more easily.

5c: [0, 0] OK, so now I’m pretty sure it the choice of terminology/labels that makes the difference, because this one is very confusing for me trying to keep track of which elements are in which sets.

Writeup Clarity Style

1a 2 1

1b 1 1

1c -2 -1

2a 0 -2

2b 2 1

2c 1 2

3a 0 -1

3b 1 1

3c 2 1

4a 2 2

4b -2 -2

4c 1 0

5a 2 2

5b 0 -1

5c 1 0

Some additional remarks

(1) In a certain sense, none of the proofs was really readable. In fact I found reading them remarkably difficult with regard to the (simple) content. [i.e. my rating for clarity is not absolut but only relative]

I think the reasons are

– to much math within the text, i. e. no strucuture that supports/guides the reader when he is following the argument. Such structure needs space but paper is cheap. Publishers may not like that but what do they gain if no one is buying their stuff because it is unintelligible?

– to much text: symbolic notation like => (used in 4a) makes reading often much easier (if not used too often …).

– in some cases bunches of different (or better: superfluous) objects (epsilon, delta, theta, eta in 1c, 2a and 4c) makes it difficult to follow the argument. What is going on, what does this object stand for etc. – all becomes suddenly confusing.

(2) I think the general style of writing should always depend very much on the audience. If the proofs are part of a monograph for experts it might be fine to keep them short without to many frills. If it’s written for “students” ( on whatever level) i think it’s very different. My minimum requirement for a readable proof is always an introductuory paragraph like in (1a) or (4a) which explains what the proof attempts to do. This is one of the resons why my rating for them is high.

On some reason this is rarely the case in many textbooks. Even worse, such explanation are mostly given in trivial cases only. If the proof is difficult and the argument is more involved, this lenghty beast is usually shortened. And this almost always starts with removing everything that serves the understanding of the reader. No explanation (why, how, what’s the next step), many implicit assumption etc. (see e.g. the proof of the Zassenhaus Lemma in Lang’s Algebra)?

(3) I absolutly hate “slick” proofs. They hide everything (motivation, idea, etc.), explain nothing and leave the reader completly baffled. Most of the time they are pure magic and they give no clue at all why the theorem is true. My personal favourite is the proof of the implicit function theorem with the contraction lemma – it works but it gives no hint at all why. All geometric intuition is thrown out of the window.

As I see it – the only purpose of a slick proof in a textbook is to show off. Alternative views might give new insights, but only if you an expert (i.e. it’s ok for mongraphs). Everybody else, in particular someone who is trying to learn something new fights most of the time with understanding (content, derivation,proof). He has better things to do than to admire a pompous author.

Happily none of your writeups belonged to this class – I would have rated it as -3/-3.

1(a) 2; 2; (b) 0; -1; (c) -1; -2

2(a) -1; -1; (b) -1; -1; (c) 2; 2

3(a) 1; 0; (b) 0; 0; (c) 2; 2

4(a) 1; 0; (b) 2; 2; (c) 1; 0

5(a) 2; 2; (b) 2; 1; (c) 1; -1

(Incidentally, your clarity/style distinction raises interesting questions. I had drafted some thoughts on that, but you did not invite them.)

]]>1a: Clarity=1, Style=1. Too wordy for my taste. For example, I’d prefer “A \subseteq B” to “A is contained in B”, especially for such simple relations as \subseteq.

1b: Clarity=0, Style=-1. Although the meaning is clear, X hasn’t been defined before it was used. So I spent some seconds searching for its definition. The -1 for style is due to using r, r’ and R: In my opinion, variable names of variables that are “symmetric” in meaning should convey that symmetry. For example, in 1a you used a and b for the radii, which I prefer since a) it directly reminds you that a is the radius in A and b is the radius in B, and b) ‘a’ and ‘b’ are similar but different letters. In contrast, r and r’ don’t show the relation to A and B and they don’t feel as symmetric. To me r’ is some special version of r, and not something that is conceptually equal to r. Also using lower case for r and r’ and upper case for R is not my taste. I prefer using the same “type” of letters for the same objects, e.g. lower case greek letters for reals, lower case latin letters for integers, upper case latin letters for sets, etc. Also, describing at the end of the proof what we want to prove simply doesn’t help. Put the explanation at the top (like in 1a) or leave it out.

1c: Clarity=0, Style=1. I think the B_x(a) notation is easier to understand then the d(x, u) < a.

2a: Clarity=0, Style=2. I like the writing, but would again prefer the ball notation.

2b: Clarity=1, Style=2.

2c: Clarity=2, Style=-1. The writing feels bumpy, especially at the beginning (e.g. sentences 1 and 2 feel totally unconnected). Easy to understand, though.

3a: Clarity=2, Style=1. Short and precise. The last two sentences should be formulated as a motivation at the beginning instead.

3b: Clarity=2, Style=0. Sounds like the proof of someone who's just picked up the subject. E.g. it's important that (a_n) converges in X.

3c: Clarity=2, Style=2. I like this one best for 3. Intention is stated at the beginning, reasoning is clear and well written.

4a: Clarity=2, Style=1.

4b: Clarity=1, Style=-1. Same issues with variable names as 1b.

4c: Clarity=2, Style=2. This is like 4a, just written a bit better. It has more flow and manages to avoid long formulas.

5a: Clarity=2, Style=2. Short and precise.

5b: Clarity=1, Style=1. The reasoning is less fluid than in 5a.

5c: Clarity=2, Style=0. Too wordy for my taste.

I would prefer to prove those problems that only require a topological space without any references to metrics. I think the proofs are more elegant without all those epsilons 🙂

]]>1(a) Clarity 2 Style 2

1(b) Clarity 2 Style 1

1(c) Clarity 1 Style 1

2(a) Clarty 1 Style 0

2(b) Clarity 2 Style 2

2(c) Clarity 2 Style 2

3(a) Clarity 2 Style 1

3(b) Clarity 2 Style 0

3(c) Clarity 2 Style 1

4(a) Clarity -1 Style 0

4(b) Clarity -1 Style -1

4(c) Clarity 1 Style 1

(Maybe I was getting tired reading. Could be that you were getting tired writing?)

5(a) Clarity 2 Style 1

5(b) Clarity 2 Style 2

5(c) Clarity 0 Style 0

b) +1 clarity, +2 style.

c) +1 clarity, +1 style.

I liked the first answer better than the other two. Though there is a certain repetition in it that the style of the second avoids. For me a blend of the simpleness of the first with a more straightforward written of the second would be the ideal.

2 a) +2 clarity, +1 style.

b) 0 clarity, 0 style.

c) +1 clarity, +1 style.

The first seems more linear to understand than the other two. In the second the heavy use of i.e. takes a little out of the style.

3 a) +1 clarity, +2 style.

b) +1 clarity, +1 style.

c) +2 clarity, +2 style.

I found the third argument more linear than the first two. It also is the more precise among the three and it spell out in words the definitions of each property used in the steps of the proof.

4 a) 0 clarity, 0 style.

b) +1 clarity, +1 style. [I believe that there is a typo on line 3, shouldn’t it be d(g(y’),z)?]

c) +2 clarity, +2 style.

I rather two and three than one, because of their symmetry. But the notation and repeated change of variables name makes 2 a little more cumbersome than number 3.

5 a) +2 clarity, +1 style.

b) +1 clarity, 0 style.

c) +1 clarity, -1 style.

Number one is the quicker and clearer argument. I think that the third though very much explained ends up having too many words. Two actually is very similar to one, however strangely less appealing.

Remark, I guess my grades were very relative – I always started trying to have a +2 for the best among the three and going down from there. So I don’t know how they may compare in absolute to the others.

]]>a: C: -1, S: 0

b: C: 1, S: 1

c. C:-2, S; -1

I’ll stop there. It’s all hard going for me.

]]>1a. Style 2, Clear 2. I liked this. The notation won’t generalise to more than 2 sets, however, we are only asked to prove it for 2 sets so that’s OK.

1b. Style -2, Clear 0. This is like a “proper mathematician” dressed up 1a in lots of notation but it gets in the way and I stop seeing little round balls. I feel I ought to like it more than 1a but I don’t! It adds nothing to define the ball notation, if you are going to use the d(x,y) notation at all then the ball adds nothing.

1c. Style 0, Clear 0. I do like that this starts from the basic definitions, but it is so verbose. (Explaining the meaning of intersection twice is a nice touch!!) Would be good cut to half the number of lines. Also given that we have 2 sets A and B I would have used “a” and “b” rather then “eta” and “theta” – anything to keep things simple.

2a Clear -2, Style -2. I didn’t like this. Better with x and x’ instead of x and y. Also, it’s too heuristic ie it sets down the order it was thought up in. I’d like to think that this helps comprehension. Certainly the opposite is often true, that a logical, here-are-the-axioms approach can bury comprehension but here things feels thin and stretched by the punchline. The other two versions do better to introduce continuity earlier. And, the sentence about continuity is convoluted.

2b Clear 1, Style 1. (I guess that’s a typo with the 2 notations of the balls.) Not helpful to mix balls and metrics, better one or the other. But, helpful to start

by stating where the proof is headed.

2c Clear 1, Style 2.

3a Clear 1, Style 1 because the {n in N} and the definition of the limit point are a bit excessive. Clear 1 because better to define the sequence as in 3b and 3c.

3b Clear 1, style 1.

3c Clear 2, style 2. Careful and clear to me.

4a,b Clear -1, Style -1. I got lost in all the extra variables.

4c Clear 2, Style 2. Working backwards from the definition works. This is kind of crude and brutal, but it doesn’t get between me and the basic idea of continuity and I can picture it how I want. 4a,b look more elegant and helpful but I have to remember all those extra variables and my thinking energy gets mopped up in following the author in the details, less is available for thinking about the problem.

5a. Clear 0, Style -1.

5b. Clear 2, Style 2. Much better choice of variables than 5a but a bit verbose relative to 5a.

5c. Clear -2, Style -2. Awful. Verbose and bad choice of variables.

]]>(b) 2,1 Loses half a style point because the ball definition should not be necessary and another half for the lack of initial motivation.

(c) 1,1 Loses two style points because of the lack of motivation and because the failure to use ball language makes it harder to read but gets one back for consistently using greek letters for numbers and latin for points (which might have made for an improvement in (a))

2(b,c,a)=1(a,b,c) in scores (though not exactly for the same reasons) except that for 2(a) I’d change the clarity score to 0 (or maybe even -1)

3(c,b,a)=1(a,b,b) except that in 3(b) the first mention of a should make clear that it is initially only known to be an element of X so maybe the clarity score should go down to 1.5

4(c,a,b)=1(a,b,c) except that in 4(c) x should be identified as a point rather than a positive number, and it might be nicer to have space-identifying subscripts on the d’s and to use x and x’ (as in 4(b)) rather than x and y for the two points in space X, and it would look less like a proof of uniform continuity if expressed in terms of balls centered at x, f(x), and g(f(x)). In 4(a) the pattern might be clearer if the ordering of x and p matched that of f(x) and y. The wrong reference to f(y’) in 4(b) is more damaging than the errors in 1(a) and 4(c) because the multiplicity of symbols makes it harder to see what is going on, so here again I’d downgrade the clarity score.

5(b,a,c)=1(a,b,c) except that for 5(c) I’d lower both scores for the unnecessary complication of introducing yet a third name for the common value of y and z.

]]>I can’t understand in which perspective you wish us to rate, I mean, most of my Maths knowledge is by self study, so at that time I wished the authors should write each and every detail, but now after coming to grad school, I feel bored to read such clearly written proofs. Like before grad school I liked Dummit Algebra, now I like Serge Lang Algebra book.

Can you give some clarity about this, I mean, who are the perspective readers? So that I can give my ratings depending.

]]>1(a) Clarity 1. Style -2

Ignoring mispring at the beginning of the second line ( where is meant), style points lost for the clumsy repetition at the beginning of the second para, and too many “so”s. I don’t at all mind contractions in general, but “it’s” here is unnecessary when “is” would be better.

1(b) Clarity 1/2. Style 0

Style points lost for introducing idea of ‘open ball’ at the end rather than, more naturally, at the beginning (and would the initial explicit definition be needed at this stage?). But proof is natural.

1(c) Clarity 0. Style 0

Quite laborious to no gain: why avoid the nice image of the open ball?

Problem 2.

2(a) Clarity -1. Style 0

2(b) Clarity -1, Style -2.

For a start, why throw in an arrow instead of if/then? All a bit clumsy, to my eyes.

2(c) Clarity 2, Style 1

Giving a little less detail (leaving the reader to recall, if necessary, why continuity has the stated application) in fact makes the proof-idea clearer. Nice unlaboured variation “hence”/”so”/”so”/”we’ve proved that”.

Problem 3

3(a) Clarity 0, Style -1

Ugly to use for points in , and quite unnecessary to subscript .

3(b) Clarity 1, Style 0

3(c) Clarity 2, Style 1

Spot on: beautifully clear. Only loses a style point for starting three sentences in a row “Since”.

Problem 4

4(a) Clarity, 1, Style 0/1

In some ways the best, but the modern notation for function composition [at least as rendered here, though perhaps that’s not the best test] isn’t easy to read, and again I don’t like sprinkling arrows for vernacular ifs.

4(b) Clarity 0/1, Style 0

Clarity is obscured by bad style choices (I had to check back for e.g. )

4(c) Clarity 1, Style 1/2

Clear proof-idea, and the old-style notation for function composition makes this more readable. But could, as in 4(a), have avoided play with

Problem 5

5(a) Clarity 1, Style -1

This is such an easy proof, it would be difficult to do it very unclearly — but the unmnemonic choice of variables knocks off style points.

5(b) Clarity 2, Style 1

Very nice, except for a style point meanly docked for three “so”s in a row …

5(c) Clarity 0, Style 0

Laboured to the point of making things harder not easier.

]]>1(b): 1, 1

1(c): -1,-1

2(a): -1.-1

A running theme in 1(c) and 2(a): I don’t like implicit quantification of the form “we know that y in A whenever d(x,y) < eta" and "u in U whenever d(f(x),u) < eta". I prever y and u in examples like this to be more formally introduced.

2(b): -1,0

I don't like what's happening in the sentence "i.e. f(z) in B_epsilon(f(x)) subset U". Just before it, there's a "for all x in Z, [something] implies [something else]", and the "i.e." bit is only unwinding a condition implied by [something else]. A semicolon is too little to connect these things together, as I see it.

2(c): 1,1.

What I'm taking from this is that there is an interesting gap in what I like to read, versus what I want students to produce. I know metric spaces, and I like reading proofs that don't force me through the triangle inequality every time. I find the notation for B_e(x) (where e is a positive real, and x is in the metric space) very helpful, and the elementary "set calculus" of balls, when taken for granted, cleans up proofs a lot. And I like it when students use this to clean up their proofs. But at the same time, in my teaching, I tend to push students who are seeing this the first time, who do their best to avoid nested "for all" statements and bare-bones applications of metric space axioms, to be clearer about what they are saying. Instead of "So f(x) \in U", for example, I'd like "By the definition of the set f^(-1)(U), we know f(x) in U." And maybe even I'd like some of the reasoning with balls explained by passing reference to the general facts about balls being used. (Even though I would not want to teach out of a textbook that wrote proofs this way!) Students learn or at least copy from each other, and I don't want weaker students getting the impression from stronger students that proof-writing is just writing series of "right" assertions, with no justification. For the same reason, I am almost at the point where use of the word "any" (as in "for any y in B_delta(x)") is not acceptable in student writing. Even though I *only* object to it in student writing. When strong students use "any", the less strong students will blur the details between "any" meaning all, and "any" meaning just one.

3(a) 0, 0.

This answer was on the borderline of poorly written to me but I would probably let it slide in a class. Ideally, a well written answer would be one that weaker students can learn from— or at least be reminded of facts from. So if this proof could recite a general relationship of closedness to containing sequential limits (whether it's by definition, or a theorem), I would love it.

3(b) -1, -2

I seem to care less than most about the importance of good notation, but experience tells me that a student who lets "a" denote the element of X that they're about to prove is really in A, will either (1) write obviously false statements immediately after, or (2) write a list of correct statements, with no indication of why any statement on the list is true, or any indication of how the statements in the list fit logically together, except what you can infer from the linear ordering of the statements on the page, and perhaps connecting words like "So" and "Because" and "Therefore". This proof, of course, was of type (2).

3(c) 1,0

I don't like the "a_n to a, for some a in X" ordering of quantifiers and try to teach away from it wherever possible. I like letters to be introduced before any statements are made about them. But other than that I liked this proof. OK, I wouldn't have chosen "a" to denote the element of X, but I loved "Since A is a closed subset of X, it must contain all its limit points." That is what I like to read in proofs. Correct statement _with a justification_ that I can read.

4(a): -1,0. I don't know why this was so hard for me to read, but it was.

4(b): 1,0.

4(c): 1,1.

5(a): 1,1. I would have gone to 2 for the quality, only I don't like "f(y) = f(z) = x. As f is injective, …" because a student reading it might think that f(y) and f(z) being equal to x was more important than f(y) and f(z) being simply equal, for the purpose of applying the fact that f was injective. Also, rather than just showing that every element of P is an element of Q, I like a proof to say at the end, "So P is a subset of Q." Or to say somewhere in the middle, that it suffices to show that an arbitrary element of P is an element of Q.

5(b): 2,2.

This is the closest to how I would like students to write this proof. It is probably too much to ask for a little reminder line explaining the reason for "a in A intersect B, so y in f(A intersect B)." It is just by definition (both of y, and of the set f(A intersect B)). But I like it when people draw attention to when they are using definitions and not anything more complicated.

5(c): 2,1.

This proof was a little symbol-heavy for me and it had the problem mentioned in the second half of my comments to 5(a).

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