We will examine some fundamental points of harmonic analysis by looking at two familiar examples. In our investigation of one of these special cases, the classical Weierstrass approximation theorem will play an essential role. An elementary (though non-trivial) proof of this result will be given. From there we look at the Stone-Weierstrass theorem, which takes us to a more abstract setting. This generalization will be used to give a proof of the Peter-Weyl theorem, a result that underlies a substantial part of representation theory.