Can someone provide me the definition of a (finite ) $\lim (s_n)$ and how it correlates to the definition of $\limsup(s_n)$?
$\lim(s_n)=+\infty$ if $\forall M>0, \exists N=N(M)\in \Re$ s.t.$\forall n>N $ we have $s_n >M$.
I believe this is the definition for the infinite case....
Let $s_n$ be a real sequence. Let $b_n$=sup{$s_n,s_{n+1},....$} for every $n=1,2,...$. Then $\limsup(s_n)=\overline {\lim s_n}=\inf b_n=(\mathop{\mathrm{inf\,sup}}s_k)$ for $k\geq n$. To understand it better if we have a sequence $s_n=(-1)^n$ then $\limsup s_n=1$ and $\liminf s_n=-1$. Also if $\limsup s_n=s$ then this means that for every $ε>0$ the set {$n:s+ε<s_n$} is finite.