## An inequality involving sqrt(2)

Prove that for all , .

Let .

First we will show that the inequality holds for . Note that and . For , we have . So the inequality holds for . Henceforth, we will assume that .

Suppose . Note that since , ; so . Hence .

Now suppose . By the Mean Value Theorem from calculus (which we will assume to be valid), there exists an element between and such that , and hence . Now . Since is strictly increasing, we have . Since , we have .

Note that , since (for example) is irreducible over . Now . Since , is a nonzero integer. In particular, we have , so that .

Hence .

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