Question about a proof of the Cauchy Criterion for Riemann Integrability Code Answer

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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 $ngeq 1$, 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 $(P_{n},Q_{n})_{ngeq 1}$ with $||P_{n}||<delta_{n}$ for each $n$, and fix $N$ large enough that $1/N<epsilon$. Then, for any $r,sgeq N$, 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, $(S(f,P_{n}))_{ngeq 1}$ is a Cauchy sequence of Riemann Sums, so we may find a real number, $gamma_{f}$ such that $S(f,P_{n})rightarrowgamma_{f}$ as $nrightarrow +infty$

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:
$$|S(f,P)-gamma_{f}| leq |S(f,P)-S(f,P_{tilde{N}})|+|S(f,P_{tilde{N}})-gamma_{f}| < 1/tilde{N}+epsilon < 2epsilon$$
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 $(P_{n},Q_{n})_{ngeq 1}$ 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?

Answer

All correct. $mbox{} mbox{} mbox{}$

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