I was studying the following property of supremum. And I'm getting quite confused because, I'm not sure that it holds for any sequences. For example I can take $x_n = -1^n$ and $y_n = -1^{n + 1}$, so $\sup{x_n} = \sup {y_n}= 1$ and $\sup{x_n + y_n} = 0$ because $x_n + y_n = 0\ \forall n$.
2026-04-01 04:29:37.1775017777
Supremum of Sum equals Sum of Suprema
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Watch the definition of $A+B$ which is $\{x+y : x \in A, y \in B \}$.
This is a set which contains all possible sums of an element from $A$ and an element from $B$ so, for your example, if you consider your sequences as defining sets $X$ and $Y$, then $1+1$ is in $X+Y$.