Polya's "Mathematics and Plausible Reasoning" is a good source. In chapter XVI he compares how many textbooks present proofs (which he calls a "deus ex machina" approach) and how it could be done showing the discovery process. The example he uses to illustrate is: If the terms of the sequence a1, a2, a3... are non-negative real numbers not all equal 0, then sum(1, infinity) (a_1, a_2, ...a_n)^1/2 < e sum(1, infinity)a_n.
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