Limit Supremum and Infimum. Struggling the concept I am struggling to understand what is the limit supremum/infimum. I've been told that it is not the same thing as "the limit of a supremum of a set" (which makes sense since the supremum/infimum is usually a number).
I've consulted with two Analysis books, but none of them seem to be able to convey it what they are trying to say.
I got an example in my notebook that may clarify my confusion

Ex. Consider $\left \{-200,100,1,2,-1,2,-1,1,2,-1  \right \}$
Then let $v_k = \sup \left \{a_n : n \geq k \right \}$ and $\limsup_{n\to\infty} a_n= \lim_{k\to\infty} v_k=2$ and $\liminf_{n\to\infty} a_n=-200$

Can someone explain to me the reasoning (without omitting any details) for the answers? I think I got a feeling for the liminf, but not limsup

I'd like to add to Christopher A. Wong's answer that the $\liminf$ is the smallest accumulation point while the $\limsup$ is the largest one. Moreover, if you have an understanding of the $\liminf$ already, then consider $\limsup a_n = -\liminf (-a_n)$.
