Problem 345 (difficulty: 4/10)

The sequence \(\displaystyle a_n\) is monotone and it has a convergent subsequence does it imply that \(\displaystyle a_n\) is convergent?

Solution:

Yes, since we have an \(\displaystyle a_{n_k}\to a\) convergent subsequence and because of the monotonicity \(\displaystyle \forall\ n>n_k\ |a_n-a|\leq |a_{n_k}-a|\), therefore \(\displaystyle a_n\to a\).

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