How to prove that the absolute value of the second derivative of J_0(x) is bounded?

Question

$\lvert J_0^{''}(x)\rvert < \dfrac {1} {2}$

How do I prove the above inequation?

From the above graphic it is clear that $\lvert J_0^{''}(0)\rvert=0.5$ is the maximum that the function can have. But how do I prove it?