US implies T1

Statement
Any US-space (i.e., a space where every convergent sequence has a unique limit) is a T1 space (i.e., every point is closed).

Related facts

 * T1 not implies US
 * Hausdorff implies US
 * Hausdorff implies KC
 * US not implies Hausdorff

Proof
Given: A US-space $$X$$.

To prove: $$X$$ is a $$T_1$$-space.

Proof: We need to show that for every point $$x \in X$$, the singleton subset $$\{ x \}$$ is closed. For this, take the sequence $$\{ x,x,x, \dots \}$$. By the definition of limit, $$x$$ is a limit of the sequence, and hence, since the space is US, it is the only limit. Thus, for any point $$y \ne x$$, there exists an open neighborhood of $$y$$ not containing $$x$$. Thus, $$\{ x \}$$ is closed.