(related to Corollary: Continuous Real Functions on Closed Intervals are Bounded)

- By hypothesis, $[a,b]$ is a closed real interval and $f:[a,b]\to\mathbb R$ is a continuous real function.
- Since on closed intervals continuous functions take a maximum and a minimum, so does $f$.
- Let $m:=\min\{f(x)\mid x\in[a,b]\}$ and $M:=\max\{f(x)\mid x\in[a,b]\}.$
- Let $K\in\mathbb R$ be a real number such that $K > |M|\ge |m|.$
- By the choice of $K,$ we have $|f(x)|\le K$ for all $x\in[a,b].$
- This means that $f$ is bounded on $[a,b].$∎

**Forster Otto**: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983