I am trying to prove the following lemma:

For any bounded function $\displaystyle f$ on $\displaystyle [a,b]$, it is always the case that $\displaystyle U(f)\geq\\L(f)$, where $\displaystyle U(f)=inf\{U(f,P):P\in{Q}\}$ and $\displaystyle L(f)=sup\{L(f,P):P\in{Q}\}$. For clarification, $\displaystyle Q$ is the collection of all possible partitions of the interval $\displaystyle [a,b]$.

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