Why do we use the subsequence definition of the limit superior?

Question

There are two commonly used definitions of the limit superior of the sequence $a_n$ indexed by the natural numbers:

1. $\lim \sup a_n = \lim_{n \to \infty} (\sup_{m \geq n} a_m)$
2. $\lim \sup a_n = \sup E$, where $E$ is the set of all limits of infinite subsequences of $a_n$.

In my experience, the first definition is not only easier to understand, but also easier to use for actually proving most statements involving the lim sup. However, some textbooks use the second definition. This has led me to suspect that there is some merit to using the second definition rather than the first: maybe there are some statements that are easier to prove using the second definition, or maybe it generalizes better in some setting. However, I'm not able to see what it is.

What are the reasons why an educator or textbook author might choose to use the second definition of the lim sup (treating the first definition as a theorem) rather than the other way around?

Answer 1

My preference goes to the first definition too. However, note that with the second definition (and with the similar one for $\liminf$), it becomes trivial to prove that:

1. $\limsup a_n\geqslant\liminf a_n$;
2. if $\limsup a_n>\liminf a_n$, then the sequence $(a_n)_{n\in\mathbb N}$ doesn't converge.

And it is now rather easy to prove that $(a_n)_{n\in\mathbb N}$ converges if and only if $\limsup a_n=\liminf a_n$.