The goal is to show that if

a,b)\rightarrow \mathbb{R}" alt="f

a,b)\rightarrow \mathbb{R}" /> is uniformly continuous, then

is bounded.

Here is my attempt...

Proof: Let

a,b)\rightarrow \mathbb{R}" alt="f

a,b)\rightarrow \mathbb{R}" /> be uniformly continuous. Then, by definition, given any

there exists a

such that for all

. Thus, f is continuous on

. Therefore, since

is closed and bounded, and

is contiuous on

, f is bounded on [x,y].

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