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) Let α be a fixed positive real number, α > 0. For a sequence {xn},...

) Let α be a fixed positive real number, α > 0. For a sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that {xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all n). (b) Prove that {xn} is bounded from below. (Hint: use proof by induction to show xn > √ α for all n) (c) Show that limn→∞ xn = √ α

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