# Prove that different definitions of big-Oh with n>=1 or n>N are equivalent

I am coming across two slightly different definitions of big-oh and need to prove that they are equivalent to each other:

Definition 1: f(n) = O(g(n)) if there exists constants c and N such that f(n) ≤ c g(n) for all n > N.

Definition 2: f(n) = O(g(n)) if there exists a constant c such f(n) ≤ c g(n) for all n≥1.

Intuitively I know that if we choose c large enough we can get rid of N like in definition 2. But how to prove that if definition 1 implies definition 2, and vice versa.

Assume g(n)>0 for all values of n (I need to prove this equality when it holds).

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 $$f(n)< cg(n)$$ holds. If it does, then continue on. If it doesn’t, we can always find a $$c’$$ such that $$f(n) holds for that particular $$n$$. $$c’>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.