Two properties about Bessel function

Question

Let $J_\nu(x)$ be the Bessel function of the first kind.

1. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$.

2. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$.

I think the above two properties of Bessel function is correct. For 1, It is easy to prove when $\nu$ is an integer. However, I have no idea of proving it for any $\nu$.