Consider the sequence
Does it have a unique cluster point?
The question is:
Let the sequence be in that has a unique cluster point, does that implies the sequence converges?
I was looking up the theorems, and I believe it is false since we did not say that the sequence is bounded. But are there any examples for that?
Yes, 0 is the unique cluster point, since for any natural N, I can find all even number of n greater than or equal N such that .
And this sequence doesn't converge, since if I let , let , then whenever , I would have when n is odd.
Is my argument proper? Thank you.