Suppose I have an element $x\colon\mathbb{N}\to\mathbb{R}$ in $\ell^\infty$. Fix $i\in\mathbb N$ and $x_m(i)$ converges to some $x(i)\in\mathbb R$ Let $N\in\mathbb N$. When does $\lim_{m\to\infty}\sup_{1\le i\le N}|x (i)-x_m(i)|=\sup_{1\le i\le N}\lim_{m\to\infty}|x (i)-x_m(i)|$ hold?
When interchanging limit and supremum over a finite set is allowed
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real-analysis
sequences-and-series
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0Sorry, I didn't see that $i$ was fixed. First, we have to be sure that all the terms in the equality exist. – 2012-09-20
1 Answers
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Your right-hand side is equal to zero. For the left hand side, $ 0\leq\lim_{m\to\infty}\sup_{1\leq i\leq N}|x(i)-x_m(i)| =\lim_{m\to\infty}\max_{1\leq i\leq N}|x(i)-x_m(i)| \leq\lim_{m\to\infty}\sum_{i=1}^N|x(i)-x_m(i)|\\ =\sum_{i=1}^N\lim_{m\to\infty}|x(i)-x_m(i)| =\sum_{i=1}^N0=0. $ So the left-hand side is also zero.