Help me prove this inequality: $$|a_1b_1+a_2b_2+\cdots+a_nb_n|\leq 1$$ if $$\begin{align*} a_1^2+a_2^2+\cdots+a_n^2=1, \\ b_1^2+b_2^2+\cdots+b_n^2=1.\end{align*}$$
Prove this inequality: $|a_1b_1+a_2b_2+\cdots+ a_nb_n|\leq 1$ for two normalised vectors
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calculus
inequality
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1Is it a pen slip of $|a_1b_1+\cdots+a_nb_n|\le1$ by [Cauchy-Schwarz inequality](http://en.wikipedia.org/wiki/Cauchy–Schwarz_inequality)? – 2012-07-15