Every bounded sequence in R has a convergent subsequence

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  1. Michael Penn
  2. math
  3. mathematics
  4. number theory
  5. abstract algebra
  6. calculus
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Комментарии
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The key thing here is that the real numbers are complete.
Intuitively, this means there are no gaps in the real numbers, unlike the rational numbers, which are full of gaps (e.g. there is a gap, or missing number, between all positive rational numbers x for which x²<2 and those for which x²≥2).
In fact, the real numbers are the completion of the rational numbers (i.e. the rational numbers with the gaps filled in; in the example above, √2 fills the gap).
The converge of Cauchy sequences is one way to characterise this completeness, and the nested sequence theorem is another. Yet another is the monotone convergence theorem.

MichaelRothwell
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Another way is to see that there must exist a monotonic subsequence.

Noam_.Menashe
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Would this proof still work for a sequence of random numbers?
This is a different video for me: one that I will need to watch over and over….

edwardlulofs
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Let's look at a proof of trombone sounds from Charlie Brown when grown-ups are talking.

britney
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You are an awesome professor thanks Michael

saadbenalla
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It's cool having had seen both this approach and the monotone convergence theorem approach in 2 of my analysis courses. While this approach is more intuitive, I still think I prefer just showing that you either have a subsequence that converges to the liminf or limsup (at the cost of coming across more esoteric smh)

TrinoElrich
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You might as well be talking to an ant here, as I didn’t understand a single word you said.

anakinlowground
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Magnet and light saber axis rings for spectrum?

ruffifuffler