Suppose that we want to find an asymptotic upper bound for a recurrence relation: $T(n)=aT \left ( \frac{n}{b}\right)+f(n)$ , $T(n)=c, \text{ when } n \leq n_0$, using the following method:
We choose a specific function $f(n)$ and we try to show that for an appropriate $c>0$ and an appropriate $n_0 \in \mathbb{N}$, it stands that $T(n) \leq c f(n)$. We suppose that $T(k) \leq c f(k), \forall k<n$ and we try to show that it stands for $n$.
Is the substitution method a good way to find a function, in order to apply the method I am asked to use?
P.S. We cannot use the master theorem.