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).