Net in a Metric Space

Question

I'm self-studying Willard's General Topology. I'm stumped on exercise 11A(3), which reads:

Let $M$ be any metric space. A mapping $P(\alpha) = x_\alpha$ of $\Omega_0$ into $M$ will be a net. Show that $x_\alpha \longrightarrow x$ in $M$ if and only if $x_\alpha$ is eventually equal to $x$.

For reference, $\omega_1$ is the first uncountable ordinal and $\Omega_0$ is the set of ordinals less than $\omega_1$.

I'm not looking for a solution; I'd like to solve this problem myself. However, I'd appreciate any hints or tips you can provide.

Many thanks!

Doug

Answer 1

HINT: If the net converges to $x$, then for each $n<\omega$ there is an $\alpha_n<\omega_1$ such that $x_\alpha\in B(x,2^{-n})$ whenever $\alpha_n\le\alpha<\omega_1$. I’ve left an additional hint in the spoiler-protected box below.

Now consider $\sup_{n<\omega}\alpha_n$ and $\bigcap_{n<\omega}B(x,2^{-n})$.