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I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of the proof of Theorem 1.6.5 (Completeness of \(\displaystyle \mathbb{R}^n\)) ...

Duistermaat and Kolk"s Theorem 1.6.5 and its proof (including the preceding notes on Cauchy Sequences) read as follows:View attachment 7713

In the above proof of Theorem 1.6.5 we read the following:

" ... ... But then the Cauchy property implies that the whole sequence converges to the same limit. ... ... "D&K don't seem to prove this statement anywhere ... presumably they think the proof is obvious ... but I cannot see exactly why it must be true ...

Can someone please demonstrate formally and rigorously that if a subsequence of a Cauchy sequence converges to a

limit then the whole sequence converges to the same limit. ... ...Peter