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Prove that 45< x_1000 <45.1

Source:

9/21/2010
Let x0=5x_0 = 5 and xn+1=xn+1xn (n=0,1,2,…)x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots ). Prove that 45<x1000<45.1.45 < x_{1000} < 45. 1.
integrationalgebraInequalityapproximationSequencerecurrence relationIMO Shortlist