Talk:Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 3

I assume you don't expect me to do anything for the time being. Ivar Sand (talk) 11:01, 28 November 2013 (UTC)


 * That is correct. We'll get to it eventually. &mdash; Lord_Farin (talk) 11:03, 28 November 2013 (UTC)

Background for the proof
Once a friend of mine said that a Cauchy sequence convergences for complete spaces. Moreover, he said that a Cauchy sequence converges for the reals because the space of real numbers is complete. I myself knew nothing about complete spaces since I had only knowledge of elementary real analysis.

However, I felt that his reference to complete spaces should be unnecessary, in other words, that there should be no mandatory logical link between the Cauchy criterion and complete spaces or other advanced concepts. So, I started constructing a proof that the Cauchy criterion and the standard convergence criterion are equivalent using only elementary real analysis. To prove that a convergent sequence is Cauchy is easy, as is well known. The challenging part was to prove that a Cauchy Sequence is convergent. --Ivar Sand (talk) 07:33, 27 November 2018 (EST)


 * I've been looking at this to see why it needs to be so complicated and why you cannot use the Bolzano-Weierstrass Theorem. My question is: is the sticking point the Continuum Property? --prime mover (talk) 02:29, 21 June 2019 (EDT)