Ok, let's assume the sum of all natural numbers is -1/12 < 0.

By the standard definition of a limit of the infinite series, it has to hold that the partial sums (1 + 2 + 3 + ... + n) converge to -1/12 as n goes to infinity.

Any finite sum of natural numbers is a natural number (basic definition of addition on natural numbers). By the definition of convergence (n goes to infinity), we have to get that the partial sum will eventually get arbitrarily close to -1/12, so there exists such n that

m = 1 + 2 + ... + n < 0.

So we need to have a natural number m < 0. By any (useful) definition of natural numbers (and <) you can find, you'll get that m >= 0 for every natural number m.

I have no issue with appeals to authority. They're often the best choice when arguing outside your field. But Euler was messing with divergent series and that's a big no no. There are no several hundreds years of proofs of this. I have only once encountered this at a university and the two proofs I've shown were done as a joke. No respectable mathematician thinks our mainstream definitions of limits / sums / natural numbers / real numbers / inequality produce this result.

I don't know where the harmonic series fits in here. But it's no different than the sum of all natural numbers in this context as they're both divergent.

Overall, I think the problem is with accepting definitions. When you go as low level as trying to prove n >= 0, you'll need to rely on some formal definitions. There are theories in math that deal with this. Two examples are set theory and type theory.

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

https://en.wikipedia.org/wiki/Type_theory

I go through those definitions in this article

https://medium.com/@nseverkar/does-a-b-equal-b-a-8b5549f1cf9c