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!
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.