Double series and iterated limits.

Question

Can we construct a double sequence $\{x_{ik}\}$of non-negative numbers such that $\sum_k\sum_i x_{ik}<\infty$ while $\sum_k x_{ik}=\infty $ for each $i$?

Answer 1

$x_{i1}+x_{i2}+...+x_{iN} \leq \sum\limits_{k=1}^{\infty} x_{k1}+\sum\limits_{k=1}^{\infty} x_{k2}+...+\sum\limits_{k=1}^{\infty} x_{kN} =\sum\limits_{j=1}^{N} \sum\limits_{k=1}^{\infty} x_{kj}\leq \sum\limits_{j=1}^{\infty} \sum\limits_{k=1}^{\infty} x_{kj}$.Letting $N \to \infty$ we get $x_{i1}+x_{i2}+... \leq \sum\limits_{j=1}^{\infty} \sum\limits_{k=1}^{\infty} x_{kj}< \infty$.

I have used only the definition an infinite sum here. No theorem is required for this.