The Cauchy Criterion is typically stated as follows:
$f$ is Riemann-Integrable on $[a,b]$ if and only if for every $\epsilon>0$, there is a $\delta>0$ so that for any two tagged partitions of $[a,b]$, $(P,Q),(P',Q')$ satisfying $||P||,||P'||<\delta$, we have that $|S(f,P)-S(f,P')|<\epsilon$.
Letting $\epsilon>0$, the proof for the $\Longrightarrow$ direction is clear.
Now, for $\Longleftarrow$, note that for each $n\geq 1$, we may find $\delta_{n}>0$ so that for any two tagged partitions, $(P,Q),(P',Q')$ satisfying $||P||,||P'||<\delta_{n}$, we have that $|S(f,P)-S(f,P')|<1/n$.
Next, choose a sequence of tagged partitions $(P_{n},Q_{n})_{n\geq 1}$ with $||P_{n}||<\delta_{n}$ for each $n$, and fix $N$ large enough that $1/N<\epsilon$. Then, for any $r,s\geq N$, we have $||P_{r}||,||P_{s}||<\max\{\delta_{r},\delta_{s}\}$, so it follows that
$$|S(f,P_{r})-S(f,P_{s})|<\max\{1/r,1/s\}\leq 1/N<\epsilon$$
Hence, $(S(f,P_{n}))_{n\geq 1}$ is a Cauchy sequence of Riemann Sums, so we may find a real number, $\gamma_{f}$ such that $S(f,P_{n})\rightarrow\gamma_{f}$ as $n\rightarrow +\infty$
Now, fix $\tilde{N}$ large enough that $1/\tilde{N}<\epsilon$ and $|S(f,P_{\tilde{N}})-\gamma_{f}|<\epsilon$. Then, for any tagged partition $(P,Q)$ with $||P||<\delta_{\tilde{N}}$, we have: $$|S(f,P)-\gamma_{f}| \leq |S(f,P)-S(f,P_{\tilde{N}})|+|S(f,P_{\tilde{N}})-\gamma_{f}| < 1/\tilde{N}+\epsilon < 2\epsilon$$ which shows that $f$ is Riemann Integrable.
My question is the following:
All the other proofs of this statement which I have seen on this site and elsewhere seem to view $\delta_{n+1}\leq \delta_{n}$ as a requirement for showing that the sequence of Riemann Sums over $(P_{n},Q_{n})_{n\geq 1}$ is Cauchy. I have not required this in my proof above, so my question is:
Is my argument correct without this requirement on the $\delta_{n}$'s? If not, where does it go wrong?
All correct. $\mbox{} \mbox{} \mbox{}$