I am trying to see if a recurrence relation where $f(n)$ has some constant factor $k$, e.g. $f(n)=kn$ where $0 < k < 1$, is $O(n)$. I think I reach that conclusion, but I want to double check to make sure. Given the following recurrence relation:

$$T(n)=2T(\frac{n}{2})+f(n)$$ $$T(n)=2T(\frac{n}{2})+kn$$ Since $0 < k < 1$, we can represent $kn=n^c$, where $0 < c < 1$

Therefore, this falls under the case 1 of the Master Theorem, because $a=2, b=2$, and therefore $log_b a = log_2 2 = 1 > c$.

Therefore, it's $O(n)$.

Is this correct? Is there another way to prove this?