Prove that a continuous function maps bounded sets to bounded sets

Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a continuous function, and $S \subset \mathbb{R}^n$ a bounded set. Now I want to show that $f(S)$ is bounded.

My thoughts: Since $S$ is bounded it's contained in an open ball $B(a,r)$ for some $r>0$. Then I want to show that $f(B(a,r))$ is a bounded set. I know from the continuity of $f$ that given $\varepsilon >0$ there is $\delta >0$ such that $f(B(a,\delta)) \subset B(f(a),\varepsilon)$ for any $a \in \mathbb{R^n}$. So can I just say I could pick the right $\varepsilon$ so that $\delta = r$? But I feel something is amiss here...