Hey Everyone! I figured I would start a discussion about the homework due Monday to hear some of your thoughts...

The homework is exercise 6.2.6:

Using the Cauchy Criterion for convergent sequences of real numbers (Theorem 2.6.4), supply a proof for Theorem 6.2.5. (First, define a candidate for $f(x)$, and then argue that $f_{n} \rightarrow f$ uniformly.)

So, Theorem 6.2.5 (what we need to actually prove) is:

A sequence of functions ($f_{n}$) defined on a set $A \subset \mathbb{R}$ converges uniformly on $A$ iff for every $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that $\vert f_{n}(x) - f_{m}(x) \vert < \epsilon$ for all $m,n \geq N$ and all $x \in A$.

Let's share some ideas!