4) Suppose x0​=23​,y0​=3, xn​=xn−1​+yn−1​2xn−1​yn−1​​ and yn​=xn​yn−1​​ for all n∈N. Prove that (a) xn​↓x and yn​↑y as n

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4 Suppose X0 23 Y0 3 Xn Xn 1 Yn 1 2xn 1 Yn 1 And Yn Xn Yn 1 For All N N Prove That A Xn X And Yn Y As N 1 (14.9 KiB) Viewed 33 times

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4) Suppose x0​=23​,y0​=3, xn​=xn−1​+yn−1​2xn−1​yn−1​​ and yn​=xn​yn−1​​ for all n∈N. Prove that (a) xn​↓x and yn​↑y as n→∞ for some x,y∈R; (b) x=y and 3.14155<x<3.14161. (x is actually π ) Note: If (xn​) is monotonically decreasing (resp. montonically increasing) and converges to x then we write xn​↓x (resp. xn​↑x ). Thus, proving this requires the use of the Monotone Convergence Theorem.