Proposition

Let $Y$ be a topological space with the trivial topology. Show that every sequence in $Y$ converges to every point of $Y$.

Solution

Let $\{ y_n \}$ be a sequence in $Y$. Let $y \in Y$ be given. Let $U$ be a neighborhood of $Y$. Since $Y$ has the trivial topology, $U$ is either empty or $Y$. Since $y \in U$, $U \ne \emptyset$. Therefore, $U = Y$.

This means that $\forall n \in N, y_n \in U$. Thus $y_n$ converges to $y$.