Talk:Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 4

The supremum of a sequence may not be an element of the sequence. Accordingly, $\sup S_m$ may not be an element of $S_m$. And, since $b_m := \sup S_m$, $b_m$ may not be an element of $S_m$. Therefore, $b_m$ may not be an element of $\sequence {a_n}$. It follows that $\sequence {b_n}$ may not be a subsequence of $\sequence {a_n}$. Therefore one cannot conclude as one does in the proof: "We also have from Subsequence of Real Cauchy Sequence is Cauchy that $\sequence {b_n}$ is also a Cauchy sequence in $\R$."

However, the comment above may be of no importance as the statement that $\sequence {b_n}$ be a Cauchy sequence seems never to be used in the proof. --Ivar Sand (talk) 10:52, 10 November 2022 (UTC)

I'll dig the book out in a bit and see whether I misrepresented it. --prime mover (talk) 11:08, 10 November 2022 (UTC)