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Unformatted text preview: b holds on the assumption that n is true. Taking F n +1 6 2 n +1 and F n 6 2 n together, n +1 is true. The principle of mathematical induction now guarantees that n is true for all natural numbers n > 0 so that F n 6 2 n for all natural numbers n . Remark : The proof actually shows that if n > 0 then F n < 2 n . A further inductive argument shows that if n is a natural number then (perhaps surprisingly) 5 F n = 1 + 5 2 n +1 1 5 2 n +1 . 1...
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 Fall '08
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