a problem arised when using mathematical induction

  When I was tring to work out the proof of the theorem 8.3(Dominated convergence theorem), in one step I applied the mathematical induction to show that

$$ \begin{equation} \sum_{i=1}^{\infty}\sum_{j=1}^{n}a_{i,j} =\sum_{j=1}^{n}\sum_{i=1}^{\infty}a_{i,j}.\end{equation} $$

  Then it occurs to me, that why can’t I just simply letting $n\rightarrow \infty$ and get the intended result? After thinking for a while(referring to 4 days), I maybe understood eventually. The reason is that when $n\rightarrow \infty$, the series must include infinitely many terms. And since there is not an $n$ s.t. $n+1=\infty$, the situation where the series includes infinitely many terms can’t be reached by the recursion process of the mathematical induction.