Finite-prefix extension of a big O bound ($f=O(g),g(n)\ne 0⟹{\sup }_n|f(n)/g(n)|<\infty$)

Let $f,g:\mathbb{N}\to \mathbb{R}$, with $g(n)\ne 0$ everywhere. An eventual bound $|f(n)|\le C|g(n)|$ can be made valid for all natural numbers by enlarging $C$ to the maximum of itself and the finitely many ratios $|f(n)|/|g(n)|$ before its cutoff. Conversely, a global bound is an eventual bound. Nonvanishing on the finite prefix is essential: $g(0)=0$, $f(0)=1$, and $f(n)=g(n)=1$ for $n\ge 1$ give an eventual bound but no global one. The useful principle is that finitely many finite ratios cannot destroy a big O notation.