Paper 4, Section I, E

Numbers and Sets | Part IA, 2016

Explain the meaning of the phrase least upper bound; state the least upper bound property of the real numbers. Use the least upper bound property to show that a bounded, increasing sequence of real numbers converges.

Suppose that an,bnRa_{n}, b_{n} \in \mathbb{R} and that anbn>0a_{n} \geqslant b_{n}>0 for all nn. If n=1an\sum_{n=1}^{\infty} a_{n} converges, show that n=1bn\sum_{n=1}^{\infty} b_{n} converges.

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