Prove the following for $f$ be a function that is bounded on $[a,b]$

Question

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 $[a,b]$.

Answer 1

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.

Answer 2

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 $[a,b]$.