The original proof of the Stone-Weierstrass theorem was elementary enough that Stone could [present it][1] in a lengthy article in *Mathematics Magazine*. There is a [much shorter proof][2] due to de Branges that uses the Krein-Milman theorem, the Hahn-Banach theorem, and the Riesz representation theorem. 


  [1]: https://doi.org/10.2307/3029750
  [2]: https://doi.org/10.1090/S0002-9939-1959-0113131-7