# How to show that a uniformly continuous function is bounded? [duplicate]

This was a homework assignment I was asked to do:

Let $f: (0,1) \to \mathbb{R}$ be uniformly continuous. Show that $f$ is bounded. (I.e. you have to show that there exist some $m, M \in \mathbb{R}$ such that $m \leq f(x) \leq M$ for all $x \in (0,1)$ ). I tried proving this using the definition of uniform continuity, but to no avail.