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    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)

    © BrainMass Inc. brainmass.com December 24, 2021, 10:23 pm ad1c9bdddf
    https://brainmass.com/math/real-analysis/real-analysis-question-three-proofs-470261

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    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 10:23 pm ad1c9bdddf>
    https://brainmass.com/math/real-analysis/real-analysis-question-three-proofs-470261

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