Proof that a set $X \subset M$ is a Manifold

Question

Let M be a manifold without boundary and let , $g:M\to \mathbb R$ has $0$ as a regular value.

Question:

Than the set of $x$ in $M$ with $g(x) \geq 0$ is a smooth manifold with boundary equal to $g^{-1}(0)$.

I am trying to prove this Lemma from Milnor's book 'Topology from a differentiable viewpoint(page 12).

Thank you.