Q: Prove that a bounded function $\displaystyle f$ is integrable on $\displaystyle [a,b]$ if and only if there exists a sequence of partitions $\displaystyle (P_{n})_{n=1}^{\infty}$ satisfying

$\displaystyle lim_{n\rightarrow\infty}[U(f,P_{n})-L(f,P_{n})]=0$.

Here some of my (incomplete) thoughts:

$\displaystyle (\Leftarrow)$

Let $\displaystyle \epsilon>0$. Since there exists a sequence of partitions $\displaystyle (P_{n})_{n=1}^{\infty}$ satisfying $\displaystyle lim_{n\rightarrow\infty}[U(f,P_{n})-L(f,P_{n})]=0$, there exists an $\displaystyle N$ such that, for any $\displaystyle n\geq\\N$ $\displaystyle U(f,P_{n})-L(f,P_{n})<\epsilon$. Thus, $\displaystyle f$ integrable on $\displaystyle [a,b]$.

$\displaystyle (\Rightarrow)$

For this direction I am not sure how to show there exists a sequence of partitions. Since we are assuming $\displaystyle f$ is integrable, I know that there is a partion $\displaystyle P_{\epsilon}$ of $\displaystyle [a,b]$ such that $\displaystyle U(f,P_{\epsilon})-L(f,P_{\epsilon})<\epsilon$ for any positive $\displaystyle \epsilon$.