I am trying to prove that if $\lim|s_n|=0$ then $\limsup|s_n|=0$. My proof goes as follows:
Suppose $\lim_{n \rightarrow \infty} s_n=0$, then this implies that $\forall \epsilon>0,\ \exists N$ such that $ n>N$ implies $|s_n-0|< \epsilon $. This implies that $\epsilon$ is an upper bound for the set $\{|s_N|\ |n>N\}$.
Thus, we can conclude that $0<\sup\{|s_n|\ |n>N\}<\epsilon$ where we know the supremum is greater than $0$ because every term $|s_n|$ is greater than $0$. Thus, from this we can conclude that $\forall \epsilon>0,\ \exists N$ such that $ n>N$ impliest that $|\sup\{|s_n|\ |n>N\}-0|< \epsilon $, and so we conclude that $\lim_{N\rightarrow \infty}\sup\{|s_n|\ |n>N\}=0$ $\rightarrow$ $\limsup|s_n|=0$.
Is this proof right? Thanks!