Baire Category Theorem

Theorem
The Baire Category Theorem states that every complete metric space is a Baire space.

Proof
Let $$U_n$$ be a countable collection of open dense subsets.

We want to show that the intersection $$\bigcap U_n$$ is dense.

A subset is dense if and only if every nonempty open subset intersects it.

Thus, to show that the intersection is dense, it is sufficient to show that any nonempty open set $$W$$ has a point $$x$$ in common with all of the $$U_n$$.

Since $$U_1$$ is dense, $$W$$ intersects $$U_1$$; thus, there is a point $$x_1$$ and $$r_1 > 0$$ such that:
 * $$\overline{B}(x_1, r_1) \subset W \cap U_1$$.

where $$B(x, r)$$ and $$\overline{B}(x, r)$$ denote an open ball centered at $$x$$ with radius $$r$$ and its closure, respectively.

Since $$U_n$$ are dense, in a recursive manner, we find a pair of sequences $$x_n$$ and $$r_n > 0$$ such that:
 * $$\overline{B}(x_n, r_n) \subset B(x_{n-1}, r_{n-1}) \cap U_n$$

as well as $$r_n < 1/n $$.

Since $$x_n \in B(x_m, r_m)$$ when $$n > m$$, we have that $$x_n$$ is a Cauchy Sequence, and $$x_n$$ converges to some limit $$x$$ by completeness.

For any $$n$$, by closedness:
 * $$x \in \overline{B}(x_{n+1}, r_{n+1}) \subset B(x_n, r_n)$$.

Hence, $$x \in W$$ and $$x \in U_n$$ for all $$n$$.