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?