The goal is to show that if $\displaystyle f

a,b)\rightarrow \mathbb{R}$ is uniformly continuous, then $\displaystyle f$ is bounded.

Here is my attempt...

Proof: Let $\displaystyle f

a,b)\rightarrow \mathbb{R}$ be uniformly continuous. Then, by definition, given any $\displaystyle \epsilon >0$ there exists a $\displaystyle \delta >0$ such that for all $\displaystyle x,y\in (a,b), |x-y|<\delta \Rightarrow |f(x)-f(y)|<\epsilon$. Thus, f is continuous on $\displaystyle [x,y]$. Therefore, since $\displaystyle [x,y]$ is closed and bounded, and $\displaystyle f:[x,y]\rightarrow \mathbb{C}$ is contiuous on $\displaystyle [x,y]$, f is bounded on [x,y].

I'll take any constructive criticism that you want to give me. Thanks.