Algebra and Geometry | Part IA, 2004

Show that

i=1naibi(i=1nai2)1/2(i=1nbi2)1/2\sum_{i=1}^{n} a_{i} b_{i} \leqslant\left(\sum_{i=1}^{n} a_{i}^{2}\right)^{1 / 2}\left(\sum_{i=1}^{n} b_{i}^{2}\right)^{1 / 2}

for any real numbers a1,,an,b1,,bna_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{n}. Using this inequality, show that if a\mathbf{a} and b\mathbf{b} are vectors of unit length in Rn\mathbb{R}^{n} then ab1|\mathbf{a} \cdot \mathbf{b}| \leqslant 1.

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