Let $f$ be bounded on a nondegenerate interval $[a,b]$. Prove that $f$ if integrable on $[a,b]$ if and only if $\epsilon > 0$, there exists a partition $P_{\epsilon}$ of [a,b] such that P is a refinement of the partition $P_{\epsilon}$ implies $|U(f,P) - L(f,P)| < \epsilon$.
attempt $\rightarrow$: Suppose $f$ if integrable on $[a,b]$, then by definition there is a partition $P_{\epsilon}$ of [a,b], such that $|U(f,P_{\epsilon}) - L(f,P_{\epsilon})| < \epsilon$. Then let P be any partition such that P is a refinement of $P_{\epsilon}$, so P is finer than $P_{\epsilon}$ . Thus, $|U(f,P) - L(f,P)| < \epsilon$.
converse attempt: Suppose $\epsilon > 0$, there exists a partition $P_{\epsilon}$ of [a,b] such that P is a refinement of the partition $P_{\epsilon}$ implies $|U(f,P) - L(f,P)| < \epsilon$, then by definition $f$ is integrable, thus f is integrable.
Can anyone verify my proof? Any feedback/help would be helpful. For the converse I am not sure, I think it is trivial. Thank you so much.