Every Cauchy sequence converges

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  1. Michael Penn
  2. math
  3. mathematics
  4. number theory
  5. abstract algebra
  6. calculus
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Комментарии
Автор

Nice, so the sequence must converge against the same limit as the converging sub sequence. We talked about this on your last video.

kilianklaiber
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It would be interesting to proof if every convergent sequence is (not) cauchy

nexonym
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Or are you Cauchy because you're a convergent sequence

Dravignor
Автор

Nice.

I don't think the N₃ is strictly necessary. As the índices nₖ are a subsequence of ℤ⁺, we have nₖ≥k, so k≥N₁⇒nₖ≥N₁. In other words, we can just take N₃ to be N₁.

MichaelRothwell
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What is a cauchy sequence and why was i recommended this?

That_Hateable_Furry
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Are you a convergent sequence because you're Cauchy

Dravignor
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I find this proof a bit disingenuous. It would be more fitting to compare the different axiomatic systems for the real numbers and show that the different ways to define the real numbers as "complete" or "not having a hole" are equivalent. What you've done here is showing one half of one direction of this equivalence.

Achill