Show that convergence of double sequence Suppose $f:X\rightarrow \mathbb R$ has property 
$$sup\left \{  {\sum_{a\in F}^{}} \left |f(a)  \right | \right \}< \infty$$ :F is finite subset of X.
1.Show that   $\left \{  \ a \in X : f(a)\neq 0 \right \}$ is countable set
2.If  $a_{kj}\in R$, show that  $$\sum_{k=1}^{\infty}\sum_{j=1}^{\infty}a_{kj}=\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}a_{kj}$$
I need for your help. Thank you for reading my problem

Let $S$ be the supremum of $\sum_{a\in F}|F(a)|$, taken over all finite subsets
$F \subset A$. Then for each $m \in \mathbb N$, the set
$$
    A_m =  \left \{  \ a \in X : |f(a)| \ge \frac 1m \right \}
$$
has at most $m \cdot S$ elements. Therefore 
$\left \{  \ a \in X : f(a)\neq 0 \right \} = \cup A_m$
is countable as the countable union of finite sets.
