Prove the following for $f$ be a function that is bounded on $[a,b]$ Let $f$ be a function that is bounded on $[a,b]$. suppose that there exists a sequence of partitions $(P_n)_{n=1}^{\infty}$ of $[a,b]$ for which:
$$\lim\limits_{n \to \infty}[\overline{S}(f,P_n)-\underline{S}(f,P_n)]=0$$
Prove that $f$ is integrable on $[a,b]$.
 A: To show that $f$ is Riemann integrable amounts to checking the following criterion is met:
Let $\epsilon >0$. We must find a partition $P$ of $[a,b]$ such that $U(P,f) - L(P,f) < \epsilon$.
The hypothesis implies that there exists $N_{\epsilon}$ such that $U(P_{n}, f) - L(P_{n}, f) < \epsilon$ for all $n \geq N_{\epsilon}$.
Take your $P = P_{N_{\epsilon}}$ and you are done. 
A: Here's one way. 
What does it mean for a function to be Riemann integrable?  Well we can say that given any $\epsilon>0$ there is a partition $\mathscr{P}$ of $[a,b]$ so that 
$$\overline{S}(f,\mathscr{P})-\underline{S}(f,\mathscr{P})<\epsilon$$
Since you know the limit given by the hypothesis exists and is equal to zero we have all the partitions we need!
Recalling what it means for the limit of a sequence to exist:
$x_n\to x_0$ if for all $\epsilon>0$ there is $N\in \mathbb{N}$ so that $|x_n-x_0|<\epsilon$ whenever $n\geq N$. 
The key here is to think of the sequence of partitions as a sequence of real numbers that are obtained by taking the difference between the upper and lower sums for the partition $\mathscr{P}_n$. 
Applying this and the definition of integrability to the problem:
Given any $\epsilon>0$  we know there is $N\in \mathbb{N}$ so that 
$$\overline{S}(f,\mathscr{P}_n)-\underline{S}(f,\mathscr{P}_n)<\epsilon$$
for every $n \geq N$. 
Therefore we conclude that the function $f$ is Riemann integrable on $[a,b]$. 
