Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 3F / b

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- Solution b

Solution

For every

N

, the Cauchy-Schwarz inequality gives

\sum_{n = 1}^{N} a_{n} b_{n} \leq (\sum_{n = 1}^{N} a_{n})^{1/2} (\sum_{n = 1}^{N} b_{n})^{1/2} .

The right-hand side is bounded independently of

N

. Since the summands are positive, the partial sums increase to a finite limit, proving the geometric means of two summable positive sequences result.

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