I want to show that if $f$ is a uniformly continuous real valued function on the bounded set $E \subseteq \mathbb{R}$, then $f$ is bounded on $E$.
I want to define an open cover of $E$, then say that cover holds true for the closure of $E$. Then $E$ is compact and by continuity, $f(E)$ is compact - and so $f$ is bounded on $E$, by the Heine-Borel theorem.
This is problem 4.8 in Rudin's Principles of Mathematical Analysis.
My concern is the first part: Is it true that I can define the same cover for $E$ as the cover for $E$ closure?
Thanks.