I'm having a bit of a problem with an exercise I have to make. In the exercise we are given the sequence $(s_n)$, which is a sequence of reals. Furthermore, we are given that $m=\sup\{s_n|n \geq 1\}<\infty$, and the supremum is not attained. We are to prove that $\limsup_{n\rightarrow\infty}s_n=m$.
Now if $s_n$ were an increasing sequence I would not have any trouble with believing this. But if $s_n$ for instance were decreasing, I don't see why this would be true. So say there is no lower bound, wouldn't that mean that even though there exists a supremum, that the $\limsup$ would be $-\infty$?
Any help is much appreciated!