Let $$X$$ be a subset of $$\mathbb{R}$$. Then the following two statements are equivalent:
2. Given any sequence $$(a_n)_{n=0}^{\infty}$$ of real numbers which takes values in $$X$$ (i.e., $$a_n \in X \text{ for all } n$$ ), there exists a subsequences $$(a_{n_j})_{j=0}^{\infty}$$ of the original squence, which converges to some number $$L$$ in $$X$$.