prove integrable functions by means of Holder inequality

Question

Let $a(t)$ be a positive continuously differentiable function. In addition, it is well known that $a(t)$ is bounded and $\int_{0}^{\infty}a(t)dt$ is bounded. I am goint to prove that $\int_{0}^{\infty}a(t)^mdt$ (where $m\geq1$)is integrable by means of Holder inequality. If this claim is not true, please help me to find counterexample.