So if anyone actually wants to know why this is not okay, it's because the child is using the "Algebraic Limit Theorem" without first showing that the series dog+dog+dog+... converges. This means that we cannot treat the limit (as n goes to infinity of the sum of n dogs - the usual definition of an infinite sum) as real number (of dogs) and hence subtraction and addition are not defined well.

Sorry, I just needed to get that off my chest.