 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-ii/2/b/iii/solution

For

n \geq 1

, the preceding parts give a positive denominator and a nonpositive numerator in

a_{n + 1} - a_{n} = \frac{c - a _{n}^{2}}{2 a _{n}} .

Hence

a_{n + 1} \leq a_{n} for every n \geq 1

. The restriction

n \geq 1

c > 1

, then

a_{1} = (1 + c) /2 > a_{0}

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