$L^p$ norm inequality for measurable function

Question

Let $E$ be a lebesgue measurable set of $\mathbb{R^n}$ and $f:E \rightarrow \mathbb{C}$ a measurable function.

Prove that for all $1\leq r \leq p \leq s < \infty$, it yields $$||f||_p^p \leq ||f||_r^r + ||f||_s^s$$

I came across this as an exercise on a real analysis course. I would assume this can be proven using either Hölder or Minkowski inequalities. Been searching for an answer here but wasn't able to find any so I think I'm not duplicating.

In this context $f$ is measurable if $ f^{-1}(M)$ is lebesgue measurable for all $M$ borel set in $\mathbb{C}$.

Answer 1

For any non-negative number $x$ we have $x^{p} \leq x^{r}+x^{q}$. In fact $x^{p} \leq x^{r}$ if $x \leq 1$ and $x^{p} \leq x^{q}$ if $x >1$. Hence $|f|^{p} \leq |f|^{r}+|f|^{q}$. Integrate both sides to derive the stated inequality.