Let x1,x2,... be a convergent sequence of L(S) and let x be its limit. We must prove that x belongs to L(S). This means proving that x is a limit point of S. Fix epsilon>0. Take x_n such that | x_n - x | < epsilon/2. Since x_n is a limit point of S, there is a y in S such that | y - x_n | < epsilon/2. Hence y is in the epsilon neighborhood of x (apply triangle inequality). Since epsilon was arbitrary, this proves that any neighborhood of x has a point of S, hence x is a limit point of S. Therefore x is in L(S) and L(S) is closed.