# Proving that $\limsup(x_n+y_n) = \limsup x_n + \lim y_n$ if $(y_n)$ converges

Let $$x_n$$ and $$y_n$$ be two bounded sequences. Assume that $$y_n$$ converges to some value b. I seek to prove that lim sup($$x_n$$ + $$y_n$$) = lim sup($$x_n$$ ) + b. I did prove that lim sup($$x_n$$ + $$y_n$$) $$\leq$$ lim sup($$x_n$$ ) + lim sup($$y_n$$) for any two bounded sequences, but I'm not sure how to continue, and not sure if this helps in the first place.

• Please see revision of title to understand how to encode mathematics on this site. – Did Sep 24 '18 at 18:47