$\lim \sup $ : "Infinitely Often" vs "Eventually" statements

Question

I can't get the meaning of "Infinitely Often" as well as "Eventually" statements in the following :

1) if $z > \lim \sup x_n$ , then : $x_n < z$ eventually (that is, for all sufficiently large $n$ )

2) if $z < \lim \sup x_n$ , then : $x_n > z$ infinitely often (that is, for infinitely many $n$)

Any help to get meaning of these both points? The intuition behind?

Answer 1

Consider the following subsets of $\mathbb N$: $$A = \{2n\mid n\in\mathbb N\},\quad B = \{ n + 100\mid n\in\mathbb N\}.$$ Set $A$ contains infinitely many positive integers, while set $B$ not only contains infinitely many positive integers, it contains all but finitely many positive integers, or you can say that $n\in B$ for sufficiently large $n$ (you can't say the same for $A$).

Can you see the difference and which kind of subset ($A$ or $B$) corresponds to your statements?

Answer 2

In terms of convergence of a sequence, it doesn't matter what happens at the start, just the long-term behaviour. You can change the first million terms of a sequence without any effect on its limiting behaviour. So any statement about terms of the sequence that is true except possibly in the first million terms might as well be true if you're only interested in limiting behaviour. And the same would work for any value of a million, i.e. for any statement that's true for all $n\geq m$ , where $m$ is any constant you like. This is what it means to be eventually true.

"Infinitely often" is the opposite: if a statement such as $x_n>z$ is true infinitely often then there's no $m$ after which its negation, $x_n\leq z$, is always true (otherwise $x_n>z$ would be true at most $m$ times, i.e. finitely often). So $\neg(P\text{ eventually})$ is the same as $((\neg P)\text{ infinitely often})$.

lim sup is the answer to a question about which bounds hold eventually. If $z$ is eventually an upper bound, then $x_n\leq z$ for all $n\geq m$ and some $m$. We can turn that around: for a fixed $m$ what is the smallest $z$ which is an upper bound for all $n\geq m$? Call this $z_m$, then the lim sup is the limit of this value as $m\to\infty$. So if $z$ is bigger than the limit of $z_m$, there certainly must be an $m$ for which $z_m<z$, and this gives you (1).