If $ \limsup ( a_n ) = L \in \mathbb{R} $ ( Meaning the largest partial limit of $ (a_n) $ is a finite number ) then does this mean $ ( a_n) $ is bounded from above?
I had this question in my mind for awhile and I think the answer is no but I couldn't think of a counter-example.
If $(a_{n})$ is not bounded from above, then there is a subsequence $(a_{n_{k}})$ such that $a_{n_{k}}\rightarrow\infty$, but then $\limsup a_{n}\geq\lim_{k}a_{n_{k}}=\infty$.