 Solution

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

Suppose

a_{n} \to ℓ

. With tolerance

1

, choose

N

such that

∣ a_{n} - ℓ ∣ < 1

whenever

n \geq N

∣ a_{n} ∣ \leq ∣ a_{n} - ℓ ∣ + ∣ ℓ ∣ < 1 + ∣ ℓ ∣

for these terms. Set

B = 1 + ∣ ℓ ∣ + j = 1 \sum N - 1 ∣ a_{j} ∣.

An empty sum is zero. This finite

B

satisfies

∣ a_{n} ∣ \leq B

for every

n

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