Greedy algorithm for submodular optimzation
The claim isn't true in general. Given a monotone submodular function $f$ and a positive constant $C$, let $g = f/C$. The algorithm is oblivious to the change from $f$ to $g$, and on the other hand $1 + \ln g(N) = 1 + \ln f(N) - \ln C$. In particular, by choosing $C$ large enough, you can get that the approximation ratio is any small enough number (even smaller than one or smaller than zero!).