Originally Posted by

**emakarov** The issue is the order of quantifiers in the statement that has to be disproved. The statement says that there exist *the same* $\displaystyle c$ and $\displaystyle B$ that work *for all* $\displaystyle n\ge B$.

Here you are saying that for all $\displaystyle n$ there exist *its own* $\displaystyle c$ that validates the statement. That would correspond to $\displaystyle \forall n\exists c\,f(n)\le cg(n)$. However, this is not what is claimed. Since $\displaystyle \exists c$ comes before $\displaystyle \forall n$, one has to choose $\displaystyle c$ first, and it has to work for all $\displaystyle n$.