> [!NOTE] Lemma > Let $I$ be a [[Bounded Real Interval|bounded real interval]]. Let $f:I\to \mathbb{R}$ be a [[Uniformly Continuous Real Function|uniformly continuous real function]]. Then $f$ is [[Bounded Real Function|bounded]]. **Proof**: Since $f:I\to \mathbb{R}$ is uniformly continuous, there exists $\varepsilon$ such that for all $x,y\in I,$ $|x-y|-\delta \implies |f(x)-f(y)|<1.$Thus $f$ is bounded on each interval $[x,x+\delta].$ Since $I$ is bounded it can covered by a finite number of such partitions.