Stone-Weierstrass: proof of lemma

In this lecture Roland Speicher (Saarland University) discusses and proves the Theorem of Stone-Weierstrass, which says that a unital subalgebra of C(K) which separates points of K is dense in C(K).
The lecture is divided into 8 small videos; in this fifth video, it is shown that a closed unital subalgebra contains with two (or a finite number of) functions also their maximum and their minimum; the main work is to prove that it contains with a positive function also its square root.