Simple to prove, yet with deep implications about the geometry of the complex plane, the Schwarz lemma is often neglected in favour of the bigger guns (like the Riemann mapping theorem). However, this little lemma is the cornerstone of all those more famous results.
Lemma. Let f(z) be a holomorphic function from the unit disk to itself, and suppose f(0)=0. Then |f(z)| ≤ |z| for all |z| < 1, and |f'(0)| ≤ 1. If equality holds anywhere (i.e. if |f(z)| = |z| for some z, or if |f'(0)|=1), then f is a rotation f(z) = wz, for some |w|=1.