I am trying to prove the following lemma:

For any bounded function on , it is always the case that , where and . For clarification, is the collection of all possible partitions of the interval .

This seems trivial, but I am not sure how to construct a solid proof. There are a few lemmas I am temped to use, but when I try to use them, I basically just end up restating the problem statement.

Any guidance would be great,

thanks