✔ explaining by contrapositive
Adding up any number of zeros gives zero; so, it is impossible to obtain a nonzero value by adding up just zeros.
explaining by deriving contradiction
Let $c$ be a nonzero value.
Assume that $n\in\mathbb Z_0^+$ zeros are required to additively obtain $c,$ i.e., $\underbrace{0+0+\ldots+0}=c.\\\quad\;n\text{ times}$
Then $n\times 0=c;$ so $c=0.$
Hence, $c$ is both zero and nonzero—which is absurd; our assumption must have been false; i.e., no countable number of zeros can sum to a nonzero value.
Addendum (expansion of my first comment below)
Consider a target-archery competition in which a contestant has scored $7$ points. Then the probability that they have hit the circular target is $1$ yet, given any point on it, the probability that they have hit that point is exactly $0.$ Here, the sum of uncountably many zeros turns out to be nonzero. (Measures only need to be countably additive.)