The intuitive picture
Even tho i had proven most things about limit superior and limit inferior, i was struggling in getting an intuitive and big-picture of limit superior and limit inferior of a sequence.
Then i had a conversation with a friend and we figured out some intuitive picture of lim it superior and limit inferior of a real number sequence ( let's call it $x_n$ here ).
We thought of the limit superior of $x_n$ as the "dividing line" between real numbers that have only finitely many terms (including none ) of the $x_n$ greater than them [ and infinitely many terms smaller than them ) , and real numbers that have infinitely many terms of $x_n$ greater than them.
The view is similar for limit inferior but switching "greater" by "smaller".
How it helped
With that view, other facts that i had a hard time understanding, came imediately ...
If we define a subsequential limit $c \in R$ of a real number sequence $x_n$ as a number that satisfies :
$\forall \varepsilon>0 , |x_n -c| < \varepsilon$ for infinitely many n's .
And then define a set $C_x$ = { w $\in$ R | w is a subsequential limit of $x_n$ }.
It becomes clear that sup C must be the real number that satisfies that definition of "dividing line"... Otherwise there would be a number greater than the dividing line which would have infinitely many terms above it ( which would contradict the definition ).
Similarly, we can understand why subsequences can't converge to values smaller than the limit inferior.
We also can understand why intuitively there is a subsequence of $x_n$ converging to lim sup and one converging to lim inf.
The reason is that for every $\varepsilon>0$ , (limsup $x_n$ - $\varepsilon $ , limsup $x_n$ +$\varepsilon$ ) and (lim inf $x_n$ - $\varepsilon $ , liminf $x_n$+$\varepsilon$ ) contain infinitely many elements, by definition of what the numbers above and bellow the "dividing lines" should satisfy.
Compatibility with the Infimum/Supremum of Sequence of Supremums/Infimum Definition
The problem is i'm assuming this intuitive view needs some little workaround/detail ( while still being intuitive and clear ) to be compatible with the other formal definition of limit superior.
This definition treats lim sup $x_n$ as the infimum of the sequence $\alpha_k$ = {$x_n : n>= k$}.
Firstly, the "dividing line" definition of Lim Sup doesn't uniquely define that sequence ( there are infinitely many other sequences whose elements only have finitely many terms of $x_n$ greater tham them ).
Secondly, we can't see the numbers above the dividing line ( limit superior of $x_n$ ) as the members of $\alpha_k$ and the dividing line as its infimum, because in the case where the infimum is a least element, we would be saying that the dividing line is above the dividing line.
So, basically i'm thinking if you guys can come with with a little-tweaked version of that intuitive view of limit superior and limit inferior that could solve the problems listed in 1) and 2).
Thanks in advance.