Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$. Consider the Neumann eigenvalue problem
$$ \Delta u =\lambda u \, \text{at} \, \Omega \\ \frac{\partial u}{\partial \vec{\nu}}=0 \;\, \text{at} \;\, \partial\Omega \, $$
We say that $\lambda$ is a eigenvalue of the above problem if there is $u \in W^{1,2}(\Omega)$ (sobolev space and $u$ is nonzero) which satisfies the relation:
$$\int_{\Omega} \partial u \partial v dx= \lambda \int_{\Omega} uvdx\ \text{ for all }\ v\in W^{1,2}(\Omega)$$
Show that the first eigenvalue is $0$ and the sequence of eigenvalues $\lambda_k \to \infty$ as $k \to \infty$.
Context
I had this exercise in a test of university. In the first part I had to prove that the eigenvalues are positive. But this is easy, only take $v=u$ and use the definition. Show that the first eigenvalue is $0$ is the second part. I think I have to use bounded operators theorems. Any help?