If $I$ is an indexing set for real numbers so $x_i \text{and} y_i$ are reals.
Is it true that $sup\{x_i + y_i: i\in I\}=sup\{x_i:i\in I\}+sup\{x_i:i\in I\}$?
I thought I need this statement to prove an exercise but I found that I only need the statement:
$sup\{x_i + y_i: i\in I\}\le sup\{x_i:i\in I\}+sup\{x_i:i\in I\}$ which is a straightforward proof by the definition and so the exercise is done but the question whether or not the equlity holds is not answered yet.
My question is, Does the equality always hold? If yes, Could you provide a simple proof,please? if not, Why not give a counterexample and the conditions (if any) under which the equality holds please?
For me, it's intuitively true, for example consider the special case when the sup is the maximum element.
Let $x_i$ be one when $i$ is odd and zero otherwise, let $y_i = 1-x_i$, then $\sup_i x_i = \sup_i y_i = 1$ and $\sup_i(x_i+y_i) = 1$.
The issue is that the indices at which $x_i$ approaches its $\sup$ may be different that the indices at which $y_i$ approaches its $\sup$.