How can I prove that $\mathbb{S}_{+}^n$ is a closed and convex set?

Question

$\mathbb{S}_{+}^n$ is the set of positive semidefinite (and symmetric) real matrices of size $n\times n$. I have to prove that this set is a closed convex cone. How can I do?

Answer 1

Note that $A \ge 0$ iff $x^TAx \ge 0$ for all $x$ iff $A \in \cap_x \{ B | x^T B x \ge 0\}$.

Note that for any $x$ that $\{ B | x^T B x \ge 0\}$ is a closed half space (hence convex). Since $0 \in \{ B | x^T B x \ge 0\}$ we see that it is a cone as well.