Proof of Bertrand's Postulate for $n\geq 6$
  We add a few ideas to Erd\H{o}s's proof of Bertrand's Postulate to produce one using a little calculus but requiring direct check only for $n\leq 5$ and one without using calculus and requiring direct check only for $n\leq 12$. The proofs can be presented to high school students. 
Manoj Verma
math.NT