= Solution
Interpret $\mathbb R_+$ as the strictly positive reals, as in the domain $x>0$ in question 1(e). A bound valid for every $n$ is immediately an eventual bound, by taking $m=0$ (or the least natural number under the other convention).
Conversely, suppose the eventual <big O notation> bound holds with constants $m,c$. Since $g(j)>0$, all the finitely many ratios $|f(j)|/|g(j)|$ with $j<m$ are finite. Choose
$$
C=\max\left(1,c,\left\{\frac{|f(j)|}{|g(j)|}:j<m\right\}\right),
$$
omitting the set of ratios when it is empty. For $n<m$ the definition of $C$ gives $|f(n)|\leq C|g(n)|$, and for $n\geq m$ the original bound does so. Hence \b[the two definitions are equivalent for the stated strictly positive codomain].
Strict positivity matters. If $\mathbb R_+$ were instead intended to include zero, an eventual bound need not extend to all indices: $g(0)=0$, $f(0)=1$, and $f(n)=g(n)=1$ for $n\geq1$ give a counterexample. More generally, extension is possible exactly when each initial zero of $g$ is also a zero of $f$.
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