Show that $\left\Vert f\right\Vert _{p}\leq\left\Vert f\right\Vert _{\infty}$ for all $1\leq p<\infty$

Question

My questión is simple:

Let $f\in L^{\infty}\left(\mu\right)$ whit $\mu$ a probability.

Show that $\left\Vert f\right\Vert _{p}\leq\left\Vert f\right\Vert _{\infty}$ for all $1\leq p<\infty$.

Remark: I ask this question on this website because I looking on the internet and I can not find information about it, which makes me distrust.

Answer 1

Observe that the sup norm means that $f(x) \leq ||f||_\infty$ for all $x \in X$ where presumably $X$ is your measure space and $\mu(X) = 1$. Then

$$ ||f||_p \;\; =\;\; \left ( \int_X |f|^p d\mu \right )^{1/p} \;\; \leq \;\; \left ( \int_X ||f||_\infty^p d\mu \right )^{1/p} \;\; =\;\; ||f||_\infty [\mu(X)]^{1/p} \;\; =\;\; ||f||_\infty. $$