From Spivak's Calculus,
For the theorem:
If $f$ is bounded on $[a,b]$, then $f$ is integrable on $[a,b]$ if and only if for every $\varepsilon > 0$ there is a partition $\mathcal{P}$ of $[a,b]$ such that $U( f, \mathcal{P}) - L( f, \mathcal{P}) < \varepsilon.$
Part of the proof is:
If $f$ is integrable sup${L(f, \mathcal{P})}$ $=$ inf $\{U(f, \mathcal{P})\}$.
This means that for each $\varepsilon > 0$ there are partitions \mathcal{P}', \mathcal{P}" with U(f, \mathcal{P}") - L(f, \mathcal{P}') < \varepsilon .
I am not getting this part. Could someone explain a little more about how "this means that"?