Yes it is correct. If you want a simple proof, just let be an arbitrary sequence of 0s and 1s and take the number . If you multiply that by and take the integer part, you’ll get something close to . Therefore, it’s easy to make sure that doesn’t tend to a limit. If you want not to tend to a limit, just take the previous and divide it by 2.

]]>As in, you let m tend to infinity and then n tend to infinity. I think that for a irrational number it can fail to have a limit. Is that correct?

]]>A large part of solving an unseen question (and doing mathematics in general) is figuring out what method to use or what lemmata to prove without being told these in advance, and without necessarily having confirmation at every step of the way that one is on the right track.

I’m guessing the examiners want to test this skill, though it’s hard to do this if the questions are too prescriptive. Even a statement saying which assumptions are allowed would probably be interpreted as a significant hint that the student should make use of those assumptions.

]]>This is what you get for not reading the whole post before commenting…

]]>1. Prove the Axiom of Archimedes.

2. Prove that t^n -> 0 whenever 0<=t<1 using the Axiom of Archimedes.

3. Using your result in 2, prove the following: let x be a real number in and let m,n …

It seems like a significant portion of this blogpost is dedicated to divining the mind of the examiners about allowed assumptions. Perhaps if the questions were written in a way that makes this clear everyone (that is, both you and the students) could spend more time focusing on mathematics.

]]>The emphasis on “equals” there is slightly misleading: the really important thing is that all that matters is that the closeness happens when is sufficiently large — the fact that we get equality is a little bonus that we could in theory have done without.

]]>I was being a bit brief there (as is appropriate if you’re in the middle of an exam). But the point is that we need to show that if is sufficiently large then the function is close to its limit. So if we prove that it actually *equals* the limit whenever then we’re done.

On a different note, I envy your students for having such a great teacher.

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