# Math Help - supremum

1. ## supremum

Could someone explain why $\sup\{x_n+y_n\}\leq\sup\{x_n\}+\sup\{y_n\}$?

Thanks

2. (Assuming $\sup x_n$ and $\sup y_n$ are finite) Let $x=\sup x_n$ and $y=\sup y_n$. Then we have

$x_n \le x, \, \forall n \in \mathbb{N}$

$y_n \le y, \, \forall n \in \mathbb{N}$.

Add the two together to get

$x_n+y_n \le x+y, \, \forall n \in \mathbb{N}$.

Since $x+y$ is an upper bound for $x_n+y_n$, it follows that

$\sup(x_n+y_n) \le x+y$.