Proving Riemann integrability using sequences of Riemann sums I am trying to prove the following:

Suppose $ f:[a,b]\rightarrow\mathbb{R} $ is bounded.  Then $ f $ is Riemann integrable if and only if for each sequence of marked partitions $\{P_n\}$ with $\{\mu(P_n)\}\rightarrow0$, the sequence $\{S(P_n,f)\}$ is convergent

,where $\mu(P)$ is the mesh of partition $P$ and $S(P,f)$ is the Riemann sum of $f$ over partition $P$.

My attempt at a solution:

Suppose  for each sequence of marked partitions $\{P\}$ with $\{\mu(P_n)\}\rightarrow0,$  $ \{S(P_n,f)\}$ converges.

Let $\epsilon&gt;0$ be given.  Then there is an $A\in\mathbb{R}$ and $N\in\mathbb{N}$ such that when $n&gt;N$, there exists $\delta$ such that $\mu(P_n)&lt;\delta\implies |S(P_n,f)-A|&lt;\epsilon$

Then, by the theorem provided by leo below, the existence of $A$ implies that $f$ is Riemann integrable.

Now suppose $f$ is integrable.  Then given $\epsilon&gt;0$, there exists $A\in\mathbb{R}$ such that there exists $\delta$ for which $\mu(P)&lt;\delta\implies |S(P,f)-A|&lt;\epsilon, \forall P$.

Then for each sequence each sequence of marked partitions $\{P\}$ with $\{\mu(P_n)\}\rightarrow0,$ $\mu(P_n)&lt;\delta$.

Then, $|S(P_n,f)-A|&lt;\epsilon$ which means that $\{S(P_n,f)\}$ converges to A.  Also by the theorem below, $A=\int f dt$
Your statement should be an easy corollary of the Du bois-Reymond and Darboux integration theorem. The proof of the theorem is rather cumbersome. So here is a reference: the proof can be found in Analysis by Its History by Hairer and Wanner.

And by the way, you need to define what convergence as $\{\mu(P_n)\}\to 0$ and what marked partition mean.
Your statement should be an easy corollary of the Du bois-Reymond and Darboux integration theorem. The proof of the theorem is rather cumbersome. So here is a reference: the proof can be found in Analysis by Its History by Hairer and Wanner.

And by the way, you need to define what convergence as $\{\mu(P_n)\}\to 0$ and what marked partition mean.