Question

The real sequence an is bounded but does not converge. Prove that it has two convergent subsequences with different limits.

Answer 1

Here's an example: for integer n>0, a_n=(-1)ⁿ=-1, 1, -1, 1, -1, 1, ... doesn't converge but is bounded, that is, the nth term is either -1 or 1, depending on whether n is odd or even. The sum of the series is either 0 or -1.

Answer 2

different limits