How to show that $L^p$ norm is monotone increasing? No, I am not assuming that $\mu$ is a probability measure Actually, that is a good point - I have read that the reverse inequality holds in $\ell^p$ spaces (sequence spaces)
Every bounded monotone sequence converges - Mathematics Stack Exchange If you want to prove the statement, if a sequence is monotone and bounded then it converges, the logically equivalent contrapositive would be, if a sequence is divergent then either it is not monotone or it is not bounded So, your idea would only get you halfway there You would also need to prove that divergent bounded sequences cannot be