TITLE: Understanding limsup
QUESTION [11 upvotes]: My textbook says:
$$\overline{s}_n = \sup \{a_n \mid n \geq N\}$$
and $\operatorname{limsup} \{a_n\}_{n \to \infty} = \lim_{N \to \infty} \overline{s}_N$.
Also, it says: As $N$ gets larger, the sup is taken over a smaller set, so the sequence of numbers $\{\overline{s}_n\}$ is monotone decreasing.
I don't understand why this must be the case. If we have a sequence $3-1/n$, then as $n$ gets larger, don't the sequence values (and hence the supremums in each interval) get larger?

REPLY [9 votes]: I know you already accepted an answer, but nothing does it like a picture for me.  Imagine our sequence does something like this where $n$ increases in the positive $x$ direction.
Then the limsup of this sequence would be the redline  (not the axis).

But if the sequence looks like this, 

our limsup will do this.