Is there a simple proof for $\int_a^bf^p=0\implies\int_a^bf=0$ when $p>1$?

Question

Let $f:[a,b]\to\mathbb{R}$ be a nonnegative Riemann integrable function. If $\int_a^bf^p=0$ for some $p>1$, then Hölder's inequality $$\left|\int_a^bfg\right|\leq\left(\int_a^b|f|^p\right)^{1/p}\left(\int_a^b|g|^q\right)^{1/q}$$ implies that $\int_a^bf=0$. I wonder if there is a simpler way (without using measures or Lebesgue integral) to show this without using the sledge hammer of Hölder's inequality?

Answer 1

Let $\epsilon >0$. Then $f \leq \frac {f^{p}} {\epsilon ^{p-1}} +\epsilon$ because $f \leq \frac {f^{p}} {\epsilon ^{p-1}}$ whenever $f > \epsilon$. [The inequality is obvious when $f \leq \epsilon$]. Hence $\int f \leq \epsilon (b-a)$. Since $\epsilon$ is arbitarary we get $\int f=0$.