Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. For $n=0$, we have $LHS=b_0=5$ and $RHS=4(0)=0$, thus $LHS\gt RHS$ and $P(0)$ is true.
3. Assume $P(k), k\gt1$ is true, that is $b_k\gt 4k$
4. For $n=k+1$, we have $LHS=b_{k+1}=4+b_k\gt 4+4k=4(k+1)=RHS$
5. Thus $P(k+1)$ is also true and we have proved the statement by mathematical induction.