Answer to Question #175361 in Real Analysis for Anand

Question

Give an example of a divergent sequence which has two convergent sequences. Justify your claim.

Answer 1

Consider the sequence $(−1)^n$ which is non convergent or a divergent sequence.

You can argue that it has a convergent subsequence in two ways:

(1) It is bounded, so by the Bolzano Weirstrass Theorem, it has a convergent subsequence.

(2) Observe that the subsequence, (1,1,1,1,1,1,1...) is convergent (since it is constant).