Let $x_n$ be a sequence. If $M =\limsup x_n$ then there is some subsequence $x_k$ with $M = \lim x_k$. Then $$ \left|x_k - M \right|< \epsilon \iff M - \epsilon < x_k < M+\epsilon \quad \forall k>K, $$
Then how are we saying that $x_k>M-\epsilon$ for infinitely many $k$ but $x_k < M+\epsilon$ for $k>K$?
Is my concept about the limit sup not correct? If so please help me with the correct concept about limit sup of a sequence.