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Suppose $f(m,n)$ is a double sequence in $\mathbb R$. Assume there exists $M>0$ such that $$\sum_{m=1}^{\infty} f(m,n) \le M$$ for all $n$. I wonder whether we have $$\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$$ I know maybe we can use the dominated convergence, but I do not know how to construct the dominating function? Can anyone help me?

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The proposed equality is false. For a counterexample, take $f(m,n)=1$ if $m=n$ and $f(m,n)=0$ otherwise.

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If $f(m,n)=\frac 1 n$ for $n \geq m$ and $0$ for $n <m$ then LHS is $1$ and RHS is $0$.

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