Proposition 8

In a metrizable space $X$, a point $x$ lies in the closure of a non-empty subset $A$ of $X$ if and only if there is a sequence of points of $A$ which converges to $x$.

We know already, from Chapter I, § 7, no. 3, that the condition is sufficient. To see that it is necessary, consider a countable fundamental system $(V_n)$ of neighbourhoods of $x$ such that $V_{n+1} \subset V_n$ for each $n$. If $x$ lies in the closure of $A$ then each $V_n$ meets $A$, and if $x_n$ lies in $V_n \cap A$, the sequence $(x_n)$ converges to $x$(**).

From Proposition 8 we deduce: