$f$ is integrable iff the $U(f;P)-L(f;P)< \varepsilon$ whenever mesh(P)<$\delta$?

Question

This is very close to what is shown in Rudin (Thm 6.6) but Rudin doesn't mention anything about a delta or a mesh in his theorem.

We were supposed to show this in class yesterday but the professor spent more time erasing wrong work than actually proving the theorem. I'm trying to clean up my notes but I want to make sure I'm getting it right.

The statement of the theorem we were given in class:

$f \epsilon R[a,b]$ and $f$ is bounded iff $\forall \varepsilon$ >0, $\exists \delta$ >0 such that $U(f;P)-L(f;P)< \varepsilon$ whenever $\mu (P)< \delta$.

I don't know how standard the notation is, but $\mu (P)$ is the mesh of $P$.

The question is to either: 1) Explain how Rudin's proof is really the same as what we were supposed to do in class even though the statement is different or 2) Explain how to prove this since class was a flop.

Thanks!