A geometric proof that e is irrational and a new measure of its irrationality

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We give a simple geometric proof that e is irrational, using a construction of a nested sequence of closed intervals with intersection e. The proof leads to a new measure of irrationality for e: if p and q are integers with q > 1, then |e - p/q| > 1/(S(q)+1)!, where S(q) is the smallest positive integer such that S(q)! is a multiple of q. We relate this measure for e to a known one and to the greatest prime factor of an integer. We make two conjectures and recall a theorem of Cantor that can be proved by a similar construction.