Let $f$ be a b-smooth function. Let $c$ and $n_0$ be constants such that $f(b n) \le c f(n)$ $\forall $ $n \ge n_0$. Show that $\forall $ $ i \in \mathbb{N}, f(b^i n) \le c^i f(n)$
I thought I should use induction. The property holds for $i = 1$ by definition. I assume it holds for $i = k$ and try to prove it for $i = k + 1$. Then I have
$$f(b^{k+1}n) \le c^{k+1}f(n)$$
and then I'm stuck. How should I continue from there?