Schrodinger equation subject to boundary condition implies quantization

I am in Section 2.4 of Sakurai’s Modern QM, 3rd Ed. We have just deduced the time independent Schrodinger equation for energy eigenstates $u_k(vec x)$ having eigenvalue $E_k$:

$$ -frac{hbar^2}{2m}nabla^2 u_k(vec x)+V(vec x)u_k(vec x)=E_ku_k(vec x)~~. tag{1}$$

We wish to consider bound states so we will impose an appropriate boundary condition:

$$ limlimits_{|vec x|toinfty} u_k(vec x)=0~~, tag{2} $$

with the understanding that

$$ E_k<limlimits_{|vec x|toinfty} V(vec x)~~.$$

Sakurai writes the following.

We know from the theory of PDEs that (1) subject to boundary condition (2) allows nontrivial solutions only for a discrete set of values of $E_k$.

I have worked through proving the discrete spectrum before, but I do not remember how it works. Can you tell me?