Proof: Cauchy Sequences are Convergent | Real Analysis

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We prove every Cauchy sequence converges. To do this we use the fact that Cauchy sequences are bounded, then apply the Bolzano Weierstrass theorem to get a convergent subsequence, then we use Cauchy and subsequence properties to prove the sequence converges to that same limit as the subsequence. With our previous proofs, we will have now proven a sequence converges if and only if it is Cauchy.

Proof of Bolzano-Weierstrass Theorem (coming soon):

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And with that, our Cauchy sequence lesson sequence comes to an end for now! Thanks for watching and give it a share if you want to help the channel grow, so I can make more real analysis lessons!

WrathofMath
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I’m probably missing something here, but wouldn’t log n be a counter example of that? Like isn’t it Cauchy, yet non convergent? Or so I thought at least.

henrilemoine
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Looks circular to me 😭😭 Proving cauchy sequence are convergent using Bolzano Weierstrass theorem. But proving bolzano Weierstrass theorem requires monotone convergence theorem. And the proof i saw on Wikipedia relies on cauchy sequence being convergent

quantumbossyearsago
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woah that was a bit fast, just a bit

rashpalsingh