Convergence in Indiscrete Space

Theorem
Let $\struct {S, \set {S, \O} }$ be an indiscrete space.

Let $\sequence {x_n}$ be any sequence in $S$.

Then $\sequence {x_n}$ converges to any point $x$ of $S$.

Proof
For any open set $U \subseteq S$ such that $x \in U$, we must have $U = S$.

Hence:
 * $\forall n \ge 1: x_n \in U$

The result follows from the definition of a convergent sequence in a topological space.