Why piecewise linear functions are dense in C[0,1] (Q1081046)

A sequence of subspaces \(\{X_ n\}\) in C[0,1] is said to be separating if there is a number \(c>0\) such that for each pair of disjoint closed subsets \(F_ 1\) and \(F_ 2\) in [0,1] and for all n sufficiently large, there exists \(x_ n\in X_ n\) with \(x_ n|_{F_ 1}=0\), \(x_ n|_{F_ 2}=1\), and \(\| x_ n\| \leq c\). The author proves that if \(\{X_ n\}\) is a separating sequence of subspaces in C[0,1], then there exists \(x_ n\in X_ n\) such that \(\| x-x_ n\| \to 0\). The author asserts that this theorem answers the question in the title, but this is not obvious to the reviewer.