Question about bounded subsequences Let $(x_n)$ be a sequence of real numbers satisfying the following:
Every bounded subsequence $(x_{n_j})$ of $(x_n)$ is convergent.
Prove:

*

*Is possible to $(x_n)$ have convergent subsequences but $(x_n)$ to be not convergent.

*If $(x_n)$ is bounded, then it is convergent.

Number (2) follows from the property because every sequence is a subsequence of itself. For number (1) I do not know if it is enough to find an example or if I have to prove it in general.
thank you

For first argument yes it is possible.
Suppose $((−1)^n)$ is convergent. The  every subsequence of
$((−1)^n)$ converges to the same limit. The subsequence of even terms converges to $1$, whereas the subsequence of odd terms converges to $−1$. Hence we conclude that $((−1)^n)$ is not convergent. Because the convergence of a sequence is unique.
This example also works for second argument.
