Tail of Convergent Net Converges

Theorem

Let $\struct {X, \tau}$ be a topological space.

Let $\struct {\Lambda, \preceq}$ be a directed set.

Let $x \in X$.

Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a net in $X$ converging to $x$.

Let $\lambda_0 \in \Lambda$.

Then $\family {x_\lambda}_{\lambda \succeq \lambda_0}$ converges to $x$.

Proof

Let:

$\Lambda_0 = \set {\lambda \in \Lambda : \lambda \succeq \lambda_0}$

Let $\preceq_0$ be the restriction of $\preceq$ to $\Lambda_0$.

From Tail of Directed Set is Directed, $\struct {\Lambda_0, \preceq_0}$ is directed, hence $\family {x_\lambda}_{\lambda \succeq \lambda_0}$ is a net.

Let $U$ be an open neighborhood of $x$ in $\struct {X, \tau}$.

Then there exists $\lambda^\ast \in \Lambda$ such that:

$x_\lambda \in U$ for $\lambda \succeq \lambda^\ast$.

Since $\struct {\Lambda, \preceq}$ is directed, there exists $\mu \in \Lambda$ such that $\lambda_0 \succeq \mu$ and $\lambda^\ast \succeq \mu$.

In particular, $\mu \in \Lambda_0$.

Then for $\lambda \succeq_0 \mu$, we have $x_\lambda \in U$ by transitivity.

Hence $\family {x_\lambda}_{\lambda \succeq \lambda_0}$ converges to $x$.

$\blacksquare$