Prove that if set $K\subset \mathbb{R}^k$ has the property that for every sequence $\left\{ x_n \right\}\subset K$ we can choose convergent subsequence $\left\{ x_{n_j} \right\}\rightarrow x\in K$ then $K$ is closed and bounded.
Seems very hard, but very interesting. I don't know how to approach. Can anybody help?