Let $(a_n)$ be a sequence of numbers. Show: If $(a_n)$ converges, than:
$\lim\limits \sup a_n= \lim\limits_{n \rightarrow \infty} a_n $
I can feel this is true intuitively, but I have no idea how to do formal proofs with lim sup as I've never really worked with the concept before. Can anyone give a helping hand?
Edit: Some more information: The definition we've been thaught is that the lim sup is equal to sup V if the squence has an upperbound, with V being the set of all the limit points of the sequence. If the sequence has no upperbound lim sup is +infinity, and if the sequence has an upperbound but V is empty lim sup is -infinity.
So what I thought was if (An) converges than it must only have 1 limit point:lim (An). Therefor lim sup (An) must be equal to lim(An), because V only has 1 element. But I have no idea how to write this formally...