Prove that different definitions of big-Oh with n>=1 or n>N are equivalent
Assume definition $1$ holds. Then we have identified a $c$ that works for all but finitely many $n$. For each of the remaining finitely many $n$, we can check if $f(n)&lt; cg(n)$ holds. If it does, then continue on. If it doesn’t, we can always find a $c’$ such that $f(n)&lt;c’g(n)$ holds for that particular $n$. $c’&gt;c$, so this will also hold for all the previously examined $n$. Iterate this through all the finite many $n$ and take the largest value of $c’$. Then definition $2$ holds.

The other direction is really easy. Try to think about it a bit more, because if you think about it in the right way it’ll be obvious.