Stone-Weierstrass approximation theorem, a constructive approach

The Weierstrass approximation theorem is well known in analysis. This theorem states that on a closed interval we can find a sequence of polynomials that comes closer and closer to any continuous real function. The importance of this theorem is that it is valid for both differentiable and non-differentiable functions. In this paper we review and prove this theorem in one and higher dimensions by a constructive method and then give examples. An idea - different from the one used in Walter Rudin's book "Principles of Mathematical Analysis" - enables us to prove this theorem in one and higher dimensions in the same fashion. The first chapter focuses on the one variable case which we illustrate by an example. The second chapter focuses on both the two and multiple variables cases which we also illustrate by an example. We use Mathematica 8 in constructing the examples.