I am looking for an example of a sequence of points with the following conditions.

Question

Usually if a set $\pmb K$ is compact, and every convergent subsequence of $(x_j)_{j=1}^\infty$, a sequence of points, converges to $x\in\pmb K$, then $(x_j)_{j=1}^\infty$ converges to $x$ as well.

However if $\pmb K$ is not compact, the statement is not necessarily true, and so I am now looking for an example that it is not.

To be specific, could anybody show me an sequence of points $(x_j)_{j=1}^\infty$, of which some subsequences converge to the same limit $x$, and yet $(x_j)_{j=1}^\infty$ does converge, but to a different limit.

Answer 1

$\{1,2,...\}$ is an example. Since no subsequence is convergent the hypothesis is vacuously satisfied. But the sequence is not convergent.

Another example: $a_n=n$ for $n$ even and $a_n=\frac 1 n$ for $n$ odd. All convergent subsequences converge to $0$ in this case.

You can take $K=\mathbb R$ for both.

Edit based on OP's comment below: if every subsequence of $(a_n)$ converges to the same limit then the sequence converges to that limit simply because $(a_n)$ is itself a subsequence!.

Answer 2

If a sequence converges to a point $x$ then every subsequence converges to the same point $x$ so it is impossible to have a subsequence converging to a point different from the limit of the convergent sequence.

More interesting is that if for some sequence every subsequence converges then they all converge to the same limit and that is the limit of the sequence as well.