Let $M$ be a smooth manifold with non-empty boundary.

Let $g$ be a smooth Riemannian metric on $M$. Is the following true?

For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ with non-positive scalar curvature such that $\|g-g_{\epsilon}\|_{C^0}<\epsilon$?

(Here $\|\cdot\|_{C^0}$ is the $C^0$ norm w.r.t $g$, but I guess one can use $g_{\epsilon}$ as well).

I ask whether the metrics of non-positive scalar curvature are dense in the space of metrics.

On other hand, I heard that the space of metrics with non-negative scalar curvature is closed in the uniform topology.

Where can I find references for these two facts?

(I am not sure if the boundary creates problems. I might have heard this about closed manifolds).

If it matters, I am fine with assuming that $M$ is contractible.