Absolute convergence of double series I read two theorems which both involve absolute convergence. However, I am confused whether they are equivalent of if one implies the other. The first theorem is:

If $\sum_{i,j=1}^{\infty} |a_{ij}|$ converges then
   $\sum_{i,j=1}^{\infty} a_{ij}$ converges.

and the second theorem is:

If the iterated series $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}
 |a_{ij}|$ converges (meaning that for each fixed $i \in \mathbf{N}$
   the series $\sum_{j=1}^{\infty} |a_{ij}|$ converges to some real
   number $b_i$ and the series $\sum_{i=1}^{\infty}b_i$ converges as
   well) then the iterated series $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}
 a_{ij}$ converges.

Both these theorems look very similar. Are they equivalent, or is the first one "stronger" than the second one? Can anyone shed some light on the differences/similarities between these two theorems? 
My random thoughts: If we let $s_{mn} = \sum_{i=1}^{m} \sum_{j=1}^{n} |a_{ij}|$ then the assumption of the first theorem can be restated as $\lim_{m,n \rightarrow \infty} s_{mn}$ exists and the assumption of the second theorem can be restated as $\lim_{m \rightarrow \infty} \lim_{n \rightarrow \infty} s_{mn}$ exists. So effectively, the assumption of the second theorem can be rephrased as: "If $\sum_{i,j=1}^{\infty} |a_{ij}|$ converges and $\lim_{n \rightarrow \infty} s_{mn}$ exists for all $m \in \mathbf{N}$, then the iterated series $\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}
 a_{ij}$ converges", is this line of thought correct?
 A: Convergence of a double series (to $S$) in the strictest sense is that for any $\epsilon >0$ there exists a positive integer $N$ such that for all $n,m > N$ we have
$$\left|\sum_{i=1}^m \sum_{j=1}^n a_{ij} - S \right| < \epsilon$$
For the first theorem, note that a comparison test applies to double series if the terms of the series are nonnegative.  This follows from a Cauchy criterion for double series.
Suppose the series $\sum_{i,j}|a_{ij}|$ converges. Let $b_{ij} = \max(a_{ij},0$) and $c_{ij} = \max(-a_{ij},0)$.
Since $0 \leqslant b_{ij} \leqslant |a_{ij}|$ and $0 \leqslant c_{ij} \leqslant |a_{ij}|$ we can now apply the comparison test to conclude that
$\sum_{i,j}b_{ij}$ and $\sum_{i,j}c_{ij}$ converge.
Therefore, we have convergence of 
$$\sum_{i,j}a_{ij} = \sum_{i,j}(b_{ij} - c_{ij}) = \sum_{i,j}b_{ij} - \sum_{i,j}c_{ij} $$
An almost identical argument can be used to prove the second theorem for iterated series.
That leaves your question of how are the theorems related.  One can prove the following as a consequence of the comparison test

If $\sum_{i,j} |a_{ij}|$ converges as a double series (with
  nonnegative terms), then the row and column sums $\sum_{i=1}^\infty
 |a_{ij}|$ and  $\sum_{j=1}^\infty |a_{ij}|$ are convergent.
  Furthermore the iterated sums $\sum_{i=1}^\infty \sum_{j=1}^\infty
 |a_{ij}|$ and $\sum_{j=1}^\infty \sum_{i=1}^\infty |a_{ij}|$ are
  convergent.

The converse is also true.  It follows simply because with nonnegative terms partial sums of the double series are increasing. If the iterated series converge then the partial sums of the double series must be bounded as well and, consequently, convergent.
In short, absolute convergence makes all modes of convergence equivalent.  The situation can be more problematic if we only have conditional convergence.  
