Equivalene of big O definitions (Limit Definition $\Longleftrightarrow$ Quantifier Definition)
This 2 definitions, of course, are not equivalent, when $g$ have $0$ subseuquence i.e. when $\exists n_k \in \mathbb{N}, n_{k+1}&gt;n_k , k\in \mathbb{N}$ such that $g(n_k)=0$. In this case, assuming, that $\frac{0}{0}$ have no sense as well as $\frac{a}{0}, a \ne 0$, then $\frac{f(n_k)}{g(n_k)}$ is not defined, while definition with quantifiers have no problem.
This is why, imho, first should be taken as base definition and second considered only in appropriate situations and why I count your question important.
If such subsequence does not exist, then your proof is correct.