>>11666521The step where you go from this finite sum to the infinite sum skips some steps, but is valid. The problem is the last step: you need to define what you mean when you multiply by infinity. In the normal definition, infinity is not an element of the reals, so It doesn’t make sense to multiply 0 and infinity. The closest you could get would be the limit of 0 times x as x goes to finitely, but that would still be 0, unlike what you wrote in your proof.
Even the definitions of the reals that do include infinity as an element you can do operations on (ie non standard analysis), 0 times infinity is also 0. This is because in general you want to define the reals to be a field, and it is derivable from the field axioms that anything times 0 is 0.
