Is every infinite complete metric space uncountabe?

Question

Let $X$ be an infinite complete metric space. My claim is that it is necessarily uncountable. Otherwise there exists a sequence in $X$ which exhausts the space(that is every point of the space is a point of the sequence). Since a singleton in $X$ is nowhere dense, a singleton containing a point of the sequence is nowhere dense. But then $X$ can be expressed as a union of countably many nowhere dense sets which contradicts Baire's category theorem. Is my claim correct?

Answer 1

This is incorrect. Why must a singleton be nowhere dense?

In fact, $\mathbb{Z}$ is a countably infinite complete metric space, with the usual metric from $\mathbb{R}$.