Let $x_n$ be a sequence such that $c\leq x_n\leq d$ for all but for finitely many $n$'s. Show that $\limsup x_n, \liminf x_n\in [c,d]$
I know this is true if the sequence is bounded for all $n$. But how can I prove this if there is a finite amount of $n$'s that don't hold the first inequality?