Subnet of Convergent Net Converges to Same Limit

Theorem

Let $\struct {X, \tau}$.

Let $x \in X$.

Let $\struct {\Lambda, \preceq}$ and $\struct {A, \sqsubseteq}$ be directed sets.

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

Let $\family {x_{\map \phi \alpha} }_{\alpha \mathop \in A}$ be a subnet of $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$.


Then $\family {x_{\map \phi \alpha} }_{\alpha \mathop \in A}$ converges to $x$.


Proof

Let $U$ be an open neighborhood of $x$.

Then there exists $\lambda_0 \in \Lambda$ such that:

$x_\lambda \in U$ for $\lambda \succeq \lambda_0$.

Since $\phi$ is cofinal, there exists $\alpha_0 \in A$ such that $\map \phi {\alpha_0} \succeq \lambda_0$.

Suppose that $\alpha \sqsubseteq \alpha_0$.

Since $\phi$ is increasing, we have $\map \phi \alpha \succeq \map \phi {\alpha_0}$.

Since $\succeq$ is transitive, we have $\map \phi \alpha \succeq \lambda_0$ and hence $x_{\map \phi \alpha} \in U$.

Hence $x_{\map \phi \alpha} \in U$ for $\alpha \sqsubseteq \alpha_0$.

Since $U$ was an arbitrary open neighborhood of $x$, $\family {x_{\map \phi \alpha} }_{\alpha \mathop \in A}$ converges to $x$.

$\blacksquare$

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