In the December issue of the American Mathematical Monthly, Haryono Tandra gave a very nice short proof of the chain rule. Here is a version that proof.
The Chain Rule If $f$ and $g$ are differentiable at $g(c)$ and $c$ respectively, then $f \circ g$ is differentiable at c. Moreover, $(f \circ g)'(c) = f'(g(c))g'(c)$.
Proof. There are two cases.
Case I - If $x_n \rightarrow c$, $x_n \neq c$ is a sequence such that $g(x_n) = g(c)$ for all $n \in \mathbb{N}$, then $g(x_n) - g(c) = 0$ for all $n$, which implies that $g'(c) = 0$ by the definition of derivative. Also, $f(g(x_n))-f(g(c)) = 0$ for all $n$, so $$\lim_{x_n \rightarrow c} \frac{f(g(x_n) - f(g(c))}{x_n - c} = 0 = g'(c) = f'(g(c)) g'(c).$$
Case II - If $x_n \rightarrow c$ is a sequence for which $g(x_n) \neq g(c)$ for all $n \in \mathbb{N}$, then: $$\lim_{x_n\rightarrow c} \frac{f(g(x_n))-f(g(c))}{x_n-c} = \lim_{x_n \rightarrow c} \frac{f(g(x_n))-f(g(c))}{g(x_n)-g(c)}\, \frac{g(x_n)-g(c)}{x_n-c} = f'(g(c)) g'(c).$$
In both cases the limit of the difference quotient for $f \circ g$ must be $f'(g(c))g'(c)$. If the difference quotient had any other limit under a different sequence, then by passing to a subsequence that fit the conditions of case I or II, we would have a contradiction. Therefore $f\circ g$ is differentiable with the desired derivative at $c$. $\Box$
Today we talked about compact sets, but I want to talk for a minute about open subsets of $\mathbb{R}$. Here is a useful fact:
Theorem A set $S \subseteq \mathbb{R}$ is open if and only if it is a union of open intervals.
You will understand open sets a lot better if you can prove this theorem.
Today we talked about the Repeating Decimals Theorem.
Theorem (Repeating Decimals Theorem) A decimal number is rational if and only if it has an eventually repeating decimal expansion.
We talked about needing to prove both the forward direction: that every rational number will have an eventually repeating decimal expansion, and the backwards direction: every repeating decimal is a fraction of two integers.
In order to prove the $\Rightarrow$ direction, it is sufficient to convert any fraction $\frac{n}{d}$ into something of the form $\frac{N}{10^k(1-10^m)}$ where m is the number of repeating digits. Then, all you need to prove is that every possible integer divisor d divides something of the form $10^k(1-10^m)$. That's all little tricky. It helps to break the problem into cases. First, if d and 10 are relatively prime, then you just need to show that there is a power of 10 that is equivalent to 1 modulo the denominator. This number is called the order of 10 modulo d and it is well known that it will exist.
If 10 and d aren't relatively prime, then you can split your fraction into two parts, one part is an integer divided by a power of 10 and the other part is an integer over a number relatively prime to d. Give it a try with a fraction like $\frac{1}{6}$ or $\frac{729}{52}$.
In 1734 the philosopher George Berkeley published The Analyst: A Discourse Addressed to an Infidel Mathematician. In it, he describes calculus as logically unsound:
And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?
Math is supposed to be the subject that everyone can agree on, but that was not true of calculus. Things got worse when people tried to understand more complicated applications of calculus, such as partial differential equations.
$$\frac{\partial T}{\partial t} = c^2 \frac{\partial^2 T}{\partial x^2}$$
The solutions for most PDE's (such as the heat equation) are infinite series of functions. Such infinite series raised all kinds of difficult questions about calculus in the 1800's. This led mathematicians such as Cauchy and Weierstrass to develop a rigorous theory of real analysis in the mid 1800's.