# If $\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \infty$ then $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} f_{n}(k) = \infty$.

Suppose $$\{f_{n}\}_{n=1}^{\infty}$$ be functions such that $$f_{n} : \Bbb{N} \rightarrow \Bbb{R}^{+}$$ for each $$n$$.

I was trying to prove –

If $$\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} f_{n}(k) = \infty$$ then $$\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} f_{n}(k) = \infty$$.

I can see that both the double summation series are equal by expanding the double summation.But I am trying to prove it ? any other thoughts?

Hm, as the order of summation are changed, is Uniform convergence likely to play any role here?