Working through Rudin's PMA, I found it very strange that following his terse yet rigorous treatise on metric spaces in Chapter 2, the author decided to introduce sequential limits in the typical epsilon-delta way. That is,
If for every $\epsilon > 0$, there exists an $N \in \mathbb{Z}^+$ such that $n \geq N \implies d(p, p_n) < \epsilon$, then $p \in X$ is a limit of $(p_n)$.
Compare this with the following:
A point $p \in X$ is the limit of a sequence $(p_n)$ iff every neighborhood of $p$ contains a tail of $(p_n)$.
Not only does the second offer a bit of geometric intuition and allow one to reason about less technicalities, it also generalizes nicely to improper limits, since neighborhoods of $\pm \infty$ in the extended real number line under the order topology are $(x, \infty]$, $[-\infty, x)$, respectively.
I understand why this definition would not make sense in a course where metrics/topologies are not properly introduced, but isn't it better in every way with those tools in place? Or perhaps I'm just not aware of sources that use it, in which case this question is rather pointless.
