If $\| f \|_p \leq 1$, then $|f(x)| \leq 1$ for some $x$

Question

Let $p \in [1,\infty)$ and $f:[0,1] \to \mathbb{R}$ be a measurable function such that $\|f\|_p \leq 1$. Then there is a $x\in [0,1]$ such that $|f(x)| \leq 1$.

This statement seems completely obvious, but I haven't been able to prove it. I appreciate any help.

Answer 1

If not, then $|f(x)| > 1$ for all $x \in [0,1]$. Hence $\|f\|_p > 1$, a contradiction.