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a. Use Theorem 2.4 to show that the sequence defined by
xn = 1/2xn−1 + 1/xn−1 , for n ≥ 1,
converges to√2 whenever x0 >√2.
b. Use the fact that 0 < (x0−√2)2 whenever x0 ≠√2 to show that if 0 < x0 <√2, then x1 >√2.
c. Use the results of parts (a) and (b) to show that the sequence in (a) converges to√2 whenever x0 > 0. |
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