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It's well known that the sum of the limits is the limit of the sum, if the limits exist. I mean $$ \lim_{j\rightarrow \infty} \ \sum_{i=1}^k x_{ij} = \sum_{i=1}^k \ \lim_{j\rightarrow \infty} x_{ij}. $$

Now let $x_{ij}$ be any none negative real sequence. Is the following inequality necessarily true? $$ \lim_{j\rightarrow \infty} \ \sum_{i=1}^\infty x_{ij} \leq \sum_{i=1}^\infty \ \lim_{j\rightarrow \infty} x_{ij} $$

If not, what would be a counterexample?

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The opposite inequality is always correct: $$\sum \limits_{i = 1}^\infty \liminf \limits_{j \to \infty} \; x_{i, j} \le \liminf \limits_{j \to \infty}\; \sum \limits_{i = 1}^\infty x_{i, j}$$

This is an application of Fatou's Lemma to the counting measure.

If you want to see an example where strict inequality holds, consider $x_{i, j} = I\{i = j\}$. Then $x_{i, j} \xrightarrow{j \to \infty} 0$, but $\sum \limits_{i = 1}^\infty x_{i, j} = 1$.

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