9.5 Workshop 5 (Reflection)

In the summary notes for Block 5, you’ve been given two proofs of the Cauchy-Schwarz inequality. We went through one of these proofs in the B5 Reading workshop. In the B5 Reading workshop, we also compared two proofs of the generalised AM-GM inequality: Cauchy’s original proof and Polya’s proof.

1. 1.
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   Re-read each proof and write a personal “proof summary”. A proof summary is the minimum information you would need to completely reconstruct the whole proof for yourself. E.g. it might be a brief as “induction using this algebraic identity.” It probably has more information.

2. 2.
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   Compare these four proofs. How are they similar? What are the key differences?

3. 3.
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   Compare your personal summaries. How confident are you with the material covered, where do you need more practice?

Complete both 10.27 and 10.33 (try to do this without looking at the solutions). Why has the same exercise been repeated in two different parts of the summary notes?

Compare your solutions.

This task guides you through the proof which uses the AM-GM inequality to prove the Cauchy–Schwarz inequality.

Let $a_{1},\,a_{2},\cdots,a_{n}$ and $b_{1},\,b_{2},\cdots,b_{n}$ be two sequences of real numbers. We will prove that

Define

1. 1.
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   Use the AM-GM inequality to show that for each $k=1,\ldots,n$:

   $$\frac{a_{k}b_{k}}{AB}\leq\frac{1}{2}\left(\frac{a_{k}^{2}}{A^{2}}+\frac{b_{k}^{2}}{B^{2}}\right).$$
2. 2.
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   Using part 1, show that

   $$\frac{1}{AB}\sum_{k=1}^{n}a_{k}b_{k}\leq 1.$$
3. 3.
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   Hence, prove the Cauchy–Schwarz inequality.