Prove that a continuous real-valued function on a closed interval in R is Riemann Integrable using only a given lemma Prove that a continuous real-valued function on a closed interval in R is Riemann Integrable using only the following lemma:
Lemma 1: A real-valued function $f$ on the interval $[a,b]$ is integrable on $[a,b]$ iff given any $\epsilon > 0, \exists \delta > 0$ s.t. $|S_1 - S_2| < \epsilon$ whenever $S_1$ and $S_2$ are Riemann Sums for $f$ corresponding to partitions of $[a,b]$ of width less than $\delta$.
This question may already be out there. However, I haven't been able to find it because this proof can be done with other techniques. I am only allowed to use this lemma here. Can anyone explain how this would be done using only this lemma? I'm not quite seeing it at the moment. Thanks!
 A: For the reverse implication we can construct a sequence of partitions $(P_n)_n$ and Riemann sums $(S(P_n,f))_n$ where both $\|P_n\| \to 0$ and the Riemann sums form a Cauchy sequence.
Hence, there exists $I \in \mathbb{R}$ and $N \in \mathbb{N}$  such that $S(P_n,f) \to I$ as $n \to \infty$, and for any $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that for all $n \geqslant N$ we have
$$\|P_n\| < \delta, \quad |S(P_n,f) -I| < \frac{\epsilon}{2} $$
In the above, we could have taken $\delta$ such that $|S(P_1,f) - S(P_2,f)| < \frac{\epsilon}{2}$ when $\|P_1\|, \|P_2\| < \delta$.  If $P$ is any partition where $\|P\| < \delta$, then since we already have $\|P_N\| < \delta$, it follows that
$$|S(P,f) - I| \leqslant |S(P,f) - S(P_N,f)|+ |S(P_N,f) - I| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon,$$
and $f$ must be Riemann integrable.
Construction of the sequences
There exists $\delta_1$ such that if $\|P\|, \|Q\| < \delta_1$, then $|S(P,f) - S(Q,f)| < 1$. We can assume $\delta_1 < 1$ and let $P_1$ be any partition with $\|P_1\| < \delta_1$ and $S(P_1,f)$ be any corresponding Riemann sum.
Further, there exists $\delta_2 < \min(\frac{1}{2},\delta_1)$ such that if $\|P\|, \|Q\| < \delta_2$, then $|S(P,f) - S(Q,f)| < \frac{1}{2}$. Let $P_2$ be any partition with $\|P_2\| < \min(\delta_2, \|P_1\|)$ and $S(P_2,f)$ be any corresponding Riemann sum.
Proceeding, for any $n \in \mathbb{N}$, there exists $\delta_n < \min(\frac{1}{n}, \delta_{n-1}) $ along with a partition $P_n$ and Riemann sum $S(P_n,f)$ such that if $\|P\|, \|Q\| < \delta_n$, then $|S(P,f) - S(Q,f)| < \frac{1}{n}$ and $\|P_n\| < \min( \delta_n , \|P_{n-1}\|)$.
Try to finish yourself by proving that $(S(P_n,f))$ is a Cauchy sequence.
