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

Hello Developer, Hope you guys are doing great. Today at Tutorial Guruji Official website, we are sharing the answer of Question about a proof of the Cauchy Criterion for Riemann Integrability without wasting too much if your time.

The question is published on by Tutorial Guruji team.

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?