Problem 226 (difficulty: 4/10)

Prove that if the sequence \(\displaystyle (a_n )\) has no convergent subsequence then \(\displaystyle |a_n |\to \infty .\)

Solution:

If the sequence \(\displaystyle |a_n |\not\to \infty \), then it has a bounded subsequence. By the Bolzano-Weierstrass theorem this subsequence has a convergent subsequence.

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