Example of an unbounded convergent net in a metric space

Question

Please could someone explain how we can obtain a convergent net in a metric space that is unbounded. I have considered indexing a subset in $\mathbb{R}$ by numbers in $(0,\infty)$ as I thought having an uncountable net would help with this but I’m unable to make such a unbounded net that is also convergent.

Answer 1

A convergent net $(x_i)_{i\in I}$ in a metric space is always eventually bounded: that is, there exists some $i\in I$ such that the set $\{x_j\}_{j\geq i}$ is bounded. However, the difference between nets and sequences here is that there can be infinitely many elements of $I$ that are not above $i$, and so the net as a whole can be unbounded. The terms of the net that are not above $i$ are completely irrelevant to its convergence, since convergence only depends on the eventual behavior of the net, and so they could take arbitrarily large values without affecting the convergence.

Here's a really simple example. Let $I=\mathbb{N}\cup\{\infty\}$, where $\infty$ is greater than every element of $\mathbb{N}$. Then every net $(x_i)_{i\in I}$ indexed by $I$ just converges to $x_\infty$, since $\infty$ is the greatest element of $I$. But such a net could be unbounded, since the values $x_n$ for $n\in\mathbb{N}$ could form some unbounded sequence.