Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 2 / b / v

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- Solution v

Solution

False. Choose the positive sequence

a_{n} = 1 + \frac{1}{n} .

It has limit of a sequence one, but Bernoulli's inequality gives

a_{n}^{n} = (1 + \frac{1}{n})^{n} \geq 1 + n ⟶ + \infty.

Thus

(a_{n}^{n})

is not a bounded sequence.

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