Basically I am asking if this sequence is "bounded" or not.
Consider the sequence $(n + \cos(n\pi)\sqrt{n^2 + 1})$. Does it have a subsequence that is convergent?
I think not because I tested $n = 2k$ and $n = 2k+1$. The first case $n = 2k$ tells me the sequence is unbounded, but the $n = 2k+1$ tells me that the sequence is bounded i.e. $1 - \sqrt{2} \leq b_{2k+1} \leq 0$.
I am thinking that it still doesn't have a convergent subsequence because the even terms tells me the whole sequence is unbounded.