If $f \colon [a,b] \to \mathbb{R}$ is continuous, then $\sup_{x ∈ [a,b]}\left | f(x)\right |$ is finite.
Attempt: Suppose $f\colon [a,b] \to \mathbb{R}$ is continuous, then by the Extreme value theorem, since $[a,b]$ is closed, and $f\colon [a,b] \to \mathbb{R}$ is continuous on $[a,b]$, then $f$ is bounded on $[a,b]$. Then $M =\sup_{x \in [a,b]} \left|f(x)\right|$ is finite.
I don't know if this is fine. Can anyone please help me? Any feedback/hint can help.
Notice that $$\sup_{x \in [a, b]} |f(x)| = \max\left(\sup_{x \in [a, b]} f(x), - \inf_{x \in [a, b]} f(x)\right)$$ So knowing that $f$ is bounded -- i.e., that that $\sup f$ and $\inf f$ are finite -- bounds $\sup |f(x)|$ as well.