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Proof

(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 $[a,b].$
  ∎

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References

Bibliography

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