Let $(X,d)$ be a complete metric space, and $U_k$ a countable collection of dense, open subsets. Let $x_0\in X$ and $\epsilon_0>0$ be given. We must show that there exists a $x\in \bigcap_k U_k$ such that $$d(x_0,x)<\epsilon_0.$$ Since $U_1$ is dense and open, we may choose an $\epsilon_1>0$ and an $x_1\in U_1$ such that $$d(x_0,x_1)<\frac{\epsilon_0}{2},\quad \epsilon_1<\frac{\epsilon_0}{2},$$ and such that the open ball of radius $\epsilon_1$ about $x_1$ lies entirely in $U_1$ . We then choose an $\epsilon_2>0$ and a $x_2\in U_2$ such that $$d(x_1,x_2)<\frac{\epsilon_1}{2},\quad \epsilon_2<\frac{\epsilon_1}{2},$$ and such that the open ball of radius $\epsilon_2$ about $x_2$ lies entirely in $U_2$ . We continue by induction, and construct a sequence of points $x_k\in U_k$ and positive $\epsilon_k$ such that $$d(x_{k-1},x_k)<\frac{\epsilon_{k-1}}{2},\quad \epsilon_k<\frac{\epsilon_{k-1}}{2},$$ and such that the open ball of radius $\epsilon_k$ lies entirely in $U_k$ .

By construction, for $0\leq j<k$ we have $$d(x_j,x_k) < \epsilon_j \left(\frac{1}{2} + \cdots + \frac{1}{2^{k-j}}\right) < \epsilon_j \leq \frac{\epsilon_0}{2^j}.$$ Hence the sequence $x_k, \; k=1,2,\ldots$ is Cauchy, and converges by hypothesis to some $x\in X$ . It is clear that for every $k$ we have $$d(x,x_k) \leq \epsilon_k.$$ Moreover it follows that $$d(x,x_k) \leq d(x,x_{k+1}) + d(x_{k},x_{k+1}) < \epsilon_{k+1} + \frac{\epsilon_{k}}{2},$$ and hence a fortiori $$d(x,x_k)<\epsilon_k$$ for every $k$ . By construction then, $x\in U_k$ for all $k=1,2,\ldots$ , as well. QED