3) A subsequence of a convergent sequence is convergent
3) every subsequence of a convergent sequence is also convergent and has the same limit as the whole sequence.
Thus, every convergent sequence is a Cauchy sequence.
PROPOSITION 23.1 Every convergent sequence is bounded.
A subsequence of a convergent sequence converges to the same limit.
(2) A Cauchy sequence that has a convergent subsequence is itself convergent.
That is, every sequence has a convergent subsequence.
Every bounded sequence in R has a convergent subsequence.
Every Cauchy sequence with a convergent subsequence converges
Every bounded sequence in R has a convegent subsequence.
We also recall the definition of convergent sequence.
We immediately proved that every convergent sequence is a Cauchy sequence.
Pick a convergent subsequence , by Bolzano-Weierstrass.
It is clear that the limit of a convergent sequence is unique.
It is clear that the limit of a convergent sequence is unique.
Therefore, we can extract a converging subsequence.
In particular, the projection on an affine subspace is unique.
By Bolzano-Weierstrass, there is a convergent subsequence.
By Bolzano-Weierstrass, there is a convergent subsequence.
By the Bolzano-Weierstrass Theorem, the sequence { x n } has a convergent subsequence { x n k } .
Requêtes fréquentes français :1-200, -1k, -2k, -3k, -4k, -5k, -7k, -10k, -20k, -40k, -100k, -200k, -500k, -1000k,
Requêtes fréquentes anglais :1-200, -1k, -2k, -3k, -4k, -5k, -7k, -10k, -20k, -40k, -100k, -200k, -500k, -1000k,
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