  In this survey article, we give an introduction to two methods of proof in
random matrix theory: The method of moments and the Stieltjes transform method.
We thoroughly develop these methods and apply them to show both the semicircle
law and the Marchenko-Pastur law for random matrices with independent entries.
The material is presented in a pedagogical manner and is suitable for anyone
who has followed a course in measure-theoretic probability theory.
