A short proof of the transcendence of the Mahler number
  We present short proofs of the transcendence of the Liouville and the Mahler numbers. The first proof is known for a long time, the second proof apparently appeared only in 2002-2003. The proofs are accessible to high-school students. 
Ashum Kaibkhanov and Arkadiy Skopenkov
math.NT math.HO