For an exercise, I am trying to prove that if $\liminf (x_n)=\liminf (y_n)=1$, and $\lim(x_n+y_n)=2$, then $\lim(x_n)=\lim(y_n)=1$. $x_n$ and $y_n$ are both bounded sequences as well.
I can see why it should make sense, I am just having trouble proving it. I tried using the fact that $\limsup (x_n+y_n)=2$ and then using $\limsup(x_n+y_n) \le \limsup(x_n)+\limsup(y_n)$, but this only got me $\limsup(x_n)+\limsup(y_n) \ge 2$ and I am not sure where to go from here. Any help is appreciated.
$$ 2=\limsup (x_n+y_n)\ge \limsup x_n+\liminf y_n= \limsup x_n+1 $$