Fix such that and choose such that there exists such that, whenever , the errors in approximating (i) by , (ii) by and (iii) by $\sum_{n=0}^N na_n z^{n-1}$ are all . Since

for some (complicated) polynomial , the left-hand side is for all sufficiently small . Hence

for all sufficiently small .

]]>I absolutely love this statement, and I think this admission makes this post much more beautiful than if you had presented the “trick” from the start. I try to make my students realize that slick textbook and article presentations frequently obscure the problem-solving process itself. While such education is not necessarily the point of journal articles, textbooks for real analysis and other subjects too often present “nice” proofs without discussing how on Earth a person actually thinks to do these things. I truly appreciate your attempts to motivate proofs and to minimize the use of “magic bullets” (e.g. use of Cauchy’s generalized MVT with specially chosen functions.)

Thank you for this blog. I enjoy it.

]]>My favourite proof of this uses the Riemann integral and the Fundamental Theorem of Calculus, but there are also some relatively elementary proofs available using e.g. suitable Mean Value Theorem estimates. ]]>

(i) converges pointwise to

(ii) converges pointwise to

(iii) is bounded by , for some independent of

then is differentiable and .

This applies easily to power series since the pointwise convergence and the bound can be established by the ratio test.

PROOF.

Let in your domain and .

For all we have

The first and the second terms tends to zero as , by the

hypothesis (i). The third term as well, by hypothesis (ii). By Taylor’s

theorem with Lagrange form of the remainder (the one you told us about few days

ago!), the last term is bounded by . This proves that is

differentiable at and that .

We are appealing here to a general principle, which is that if some functions converge to and their derivatives converge to , then is differentiable with . Is this general principle correct?

If you assume **uniform** convergence of the functions, this principle becomes rather usefully true, doesn’t it?

However this is presumably dealt with later (perhaps along with the Riemann integral?).