Certain property of supremum

Question

Given real numbers $x_{ni}, n\in \mathbb{N}, i\in I$, does it hold that $$\sup \bigg\{ \sum_{n\in\mathbb{N}}x_{ni}|i\in I\bigg\}=\sum_{n\in \mathbb{N}}\sup\bigg\{x_{ni}|i\in I \bigg\}?$$

Thanks in advance.

Answer 1

No. A counterexample is $I = \{ \alpha, \beta \}$, $$x_{n, \alpha} = \frac{1}{n^2}$$ $$x_{1, \beta} = \frac{5}{4}, \ \ x_{n, \beta} = 0 \ \mbox{for $n \geq 2$}$$ Then $$\sup_{i \in I} \sum_{n \in \mathbb{N}} x_{n,i}= \max \{ \frac{\pi^2}{6} , \frac{5}{4}\} = \frac{\pi^2}{6}$$ while $$\sum_{n \in \mathbb{N}} \sup_{i \in I} x_{n,i} = \frac{5}{4} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} + \frac{1}{4}$$