 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-ii/2/b/iv/solution

True. Choose

N

such that

ℓ /2 < a_{n} < 3 ℓ /2

for every

n \geq N

. All the finitely many preceding terms are also positive. Define the positive minimum and finite maximum

η = min ({ℓ /2} \cup {a_{j} : 1 \leq j < N}) > 0, B = max ({3 ℓ /2} \cup {a_{j} : 1 \leq j < N}) < \infty.

Every term lies between

η

and

B

, so every ratio is at most

B / η

. Consequently the required supremum satisfies

sup {a_{n} / a_{m} : m, n \in N} \leq B / η < \infty.

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