I was reading a proof and it said that let $q$ = lim sup $s_{n}$. Then q is the largest possible value any $s_{n}$ in a tail of the sequence of $s_{n}$ can attain. So, sup $T_{m} \leq q$ where $T_{m} = $ { $s_{n} | n \geq M$}
But as I was thinking about this and I want to say this is false. Because isn't the lim sup $s_{n}$ = inf sup $s_{n}$? So in fact lim sup $s_{n}$ would be less than any sup $T_{m}$.
Could someone clarify this for me?