23 дек. 2012 г. · This is the so called Bolzano-Weierstrass Theorem. I will prove that any sequence of real numbers has a monotone subsequence and leave the rest to you.

6 окт. 2016 г. · We can construct a subsequence that is convergent in the first coordinate by selecting every other element of the original sequence to obtain ( ...

19 янв. 2013 г. · Note: it is true that every bounded sequence contains a convergent subsequence, and furthermore, every monotonic sequence converges if and only ...

1 янв. 2017 г. · Before I ask here, I thought Bolzano-Weierstrass theorem which is every bounded sequence has convergent subsequence. And my question comes up ...

13 мар. 2016 г. · Bounded sequences don't have convergent subsequences in general. Take e.g. d(x,y)=min{|x−y|,1} on R. It's a metric that gives the standard ...

3 дек. 2017 г. · If every bounded sequence has a convergent subsequence, in particular, the limit of that convergent subsequence is an accumulation point for ...

12 июн. 2019 г. · Every sequence in a closed and bounded subset is bounded, so it has a convergent subsequence, which converges to a point in the set, because the set is closed.

21 мар. 2021 г. · Bounded sequence with subsequences all converging to the same limit means that the sequence itself converges to the same limit. 1 · Subsequences ...

15 июн. 2021 г. · If you start from a convergent subsequence, you produce infinitely many convergent subsequences just by deleting different terms.

15 июн. 2015 г. · Bounded sequence with subsequences all converging to the same limit means that the sequence itself converges to the same limit.