TITLE: Difference between limit superior & supremum of a sequence
QUESTION [8 upvotes]: Can anyone please tell what's the difference between the terms 

limit superior,   
limit inferior,
supremum,
infimum. 

They are confusing me.

REPLY [4 votes]: The supremum is in fact the least upper bound of the set or sequence. That is, of all upper bounds, it is the smallest.  All others are greater than it. Think about the open interval, say A = (0,1), in $\mathbb{R}$.  3, 75, 1.4 are all upper bounds of A, since $\forall x \in$ A, $x \le$ any of these. Of all the upper bounds of A, however, none can be smaller than 1. Thus 1 is the least upper bound, or supremum, of A.  Similarly, 0 is the greatest lower bound, or infimum, of A. 
Consider the sequence $\{ a_n\}$, where $a_n = (-1)^n$.  You can probably determine that this does not converge. There are subsequences, $\{ a_{n_k}\}$ and  $\{ a_{n_i}\}$, $n_k$ = 1,3,5... and $n_i$ = 2,4,6..., that do, to -1 and 1, respectively. So E = {-1,1} is the set of subsequential limits of $\{ a_n\}$.  The supremum, or maximum of this finite set, is 1. This is the lim sup of $\{ a_n\}$.  Similarly, lim inf = -1. 
This also works, if the subsequences do not attain the limit either, as in the sequence $\{ b_n\}$, where $b_n$ = $\frac{1}{n} + (-1)^n$. Here, lim sup = 1, but sup = $\frac{3}{2}$, lim inf = inf = -1