Can I take the limit inside the summation of a series.

Question

I was trying to solve an exercise problem of a text book, when I encountered this problem; I had to show that, for any infinite matrix of real numbers, denoted by $\{a_{j,k}\}$ such that the columns are weakly increasing and bounded and the rows are summable for each row; $$\lim_{j\to\infty}\sum_{k=1}^{\infty}a_{j,k}=\sum_{k=1}^{\infty}\lim_{j\to\infty}a_{j,k} $$ under the condition that the limit on the left hand side and the sum on the right hand side exists.

This is basically the monotone convergence lemma for series if $a_{j,k}>0$. My argument in solving the problem is as follows. Let $F(N)=\lim\limits_{j\to\infty}\sum\limits_{k=1}^{N}a_{j,k}$; and $G(N)=\sum\limits_{k=1}^N\lim\limits_{j\to\infty}a_{j,k}$. For every $N\in\mathbb{N}$ $F(N)=G(N)$, hence their limits are also the same.

It appears too trivial to be correct, any help of where I am going wrong would be appreciated. Note that I am not even using the criteria of the columns weakly increasing.

Answer 1

Replacing $a_{j,k}$ by $a_{j,k} - a_{1,k}$ if needed, we may assume WLOG that $a_{j,k} \geq 0$. Then it suffices to prove the monotone convergence theorem:

Theorem. If $(a_{j,k} : j = 1, 2, \cdots)$ is non-negative and monotone increasing for each $k$, then

$$ \lim_{j\to\infty} \sum_{k=1}^{\infty} a_{j,k} = \sum_{k=1}^{\infty} \lim_{j\to\infty} a_{j,k}. $$

To this end, we invoke the following easy-to-prove lemma:

Lemma. (1) If $A, B$ are non-empty sets and $F : A\times B \to \mathbb{R}$ is a function, then

$$ \sup_{a\in A}\sup_{b\in B} F(a,b) = \sup_{(a,b)\in A\times B} F(a, b) = \sup_{b\in B}\sup_{a\in A} F(a, b). $$

(2) If $(a_n)$ is monotone increasing, then $\lim_{n\to\infty} a_n = \sup_{n\in\mathbb{N}} a_n$.

In order to utilize this lemma, define $F : \mathbb{N} \times \mathbb{N} \to \mathbb{R}$ by

$$ F(j, N) = \sum_{k=1}^{N} a_{j,k}. $$

Then it follows that $F$ is monotone increasing in both $j$ and in $N$, hence

$$ \lim_{j\to\infty} \sum_{k=1}^{\infty} a_{j,k} = \sup_{j\in\mathbb{N}} \sup_{N\in\mathbb{N}} F(j, N) = \sup_{N\in\mathbb{N}} \sup_{j\in\mathbb{N}} F(j, N) = \sum_{k=1}^{\infty} \lim_{j\to\infty} a_{j,k}. $$