# Real Analysis Question: Three Proofs

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Let y be a positive real number. Choose x_1>sqrt(y) and let x_n+1=1/2(x_n + y/x_n), for all n>=1.

1) Prove {x_n} is monotonically decreasing and bounded.

2) Prove limit of x_n as n approaches infinity is sqrt(y)

3) Letting r_n=x_n-sqrt(y) show that r_n+1=r^2_n/2x_n < r^2_n/2sqrt(y) for all n>=1. Conclude that r_n+1<z(r_1/z)^(2^n) for all n>=1, where z=2sqrt(y)

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