TITLE: If $x_n+y_n$ is convergent, are $x_n$ and $y_n$?
QUESTION [5 upvotes]: If $x_n$ and $y_n$ are sequences such that $(x_n + y_n)$ is convergent. Does it mean that both $x_n$ and $y_n$ have to be convergent?

If it does not, please provide an example.
Thanks a lot!

REPLY [2 votes]: The sequence $x_n=n$ is divergent, and also $y_n=-n$. But $z_n=x_n+y_n=0$ is convergent. Then the answer is not, in general. The reciprocal, is of course true.  If $x_n$ and $y_n$ are both convergent then any linear combination, in particular $(x_n+y_n)$ is convergent.