Suppose $a_n \rightarrow +-\infty$ and $(b_n)$ is bounded. Show that $a_n+b_n \rightarrow +-\infty$. I tried this:
$|a_n|\rightarrow +-\infty$, so $|a_n+-\infty|<\epsilon$. It is also true that $|b_n|<M$, because $(b_n)$ was bounded. Now, if you add those together, you will get: $|a_n+-\infty|+|b_n|<\epsilon+M$ and therefor: $|a_n+b_n+-\infty|\le|a_n+-\infty|+|b_n|<\epsilon+M$. But, since M is fixed (say M was $2$), you can't get closer than $2$ to the limit (because $\epsilon$ must be negative in that case). Am I missing something?