# Interchange the order of limit and summation

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?

The proposed equality is false. For a counterexample, take $$f(m,n)=1$$ if $$m=n$$ and $$f(m,n)=0$$ otherwise.
If $$f(m,n)=\frac 1 n$$ for $$n \geq m$$ and $$0$$ for $$n then LHS is $$1$$ and RHS is $$0$$.