Characterization of Convergent Net in Metric Space

Theorem

Let $\struct {X, d}$ be a metric space.

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

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

Let $x \in X$.

Then $\family {x_\lambda}_{\lambda \in \Lambda}$ converges in $\struct {X, d}$ if and only if:

for each $\epsilon > 0$ there exists $\lambda_0 \in \Lambda$ such that $\map d {x_\lambda, x} < \epsilon$ for $\lambda \in \Lambda$ with $\lambda_0 \preceq \lambda$.

Proof

Necessary Condition

Suppose that $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$ in $\struct {X, d}$.

Let $\epsilon > 0$.

From Open Ball is Open Set, the open ball $\map {B_\epsilon} x$ with radius $\epsilon$ and center $x$.

Since $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$, there exists $\lambda_0 \in \Lambda$ such that:

$x_\lambda \in \map {B_\epsilon} x$ for $\lambda \in \Lambda$ with $\lambda_0 \preceq \lambda$.

That is:

$\map d {x_\lambda, x} < \epsilon$ for $\lambda \in \Lambda$ with $\lambda_0 \preceq \lambda$.

$\Box$

Sufficient Condition

Suppose that:

for each $\epsilon > 0$ there exists $\lambda_0 \in \Lambda$ such that $\map d {x_\lambda, x} < \epsilon$ for $\lambda \in \Lambda$ with $\lambda_0 \preceq \lambda$.

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

Since $U$ is open, there exists $\epsilon > 0$ such that:

$\map {B_\epsilon} x \subseteq U$

By hypothesis, there exists $\lambda_0 \in \Lambda$ such that $\map d {x_\lambda, x} < \epsilon$ for $\lambda \in \Lambda$ with $\lambda_0 \preceq \lambda$.

Then for $\lambda_0 \preceq \lambda$ we have $x_\lambda \in \map {B_\epsilon} x \subseteq U$.

$\blacksquare$