Inclusion in $L_p$ space

Question

I have been wondering how to prove the following statement, and would greatly appreciate your help:

If $f$ is a bounded function on $E$ that belongs to $L_{p_1}(E)$, then it belongs to $L_{p_2}(E)$ for any $p_2>p_1$.

Thank you in advance.

Answer 1

I don't see a way to get this result from Hölder's inequality, but I offer the following profound observation: if $|y|\le M$, then $$|y|^{p_2}\le M^{p_2-p_1}|y|^{p_1}$$ I hope this is sufficient for the OP.