Need help proving Cauchy Schwarz inequality ...(adsbygoogle = window.adsbygoogle || []).push({});

the first method I know is pretty easy

[itex]\displaystyle\sum_{i=1}^n (a_ix-b_i)^2 \geq 0 [/itex]

expanding this and using the discriminatant quickly establishes the inequality..

The 2nd method I know is I think a easier one , but I dont have a clue about how this notation works..

Since cauchy SHwarz inquality states..

[tex](a_1b_1+a_2b_2+...+a_nb_n)^2 \leq ((a_1)^2+(a_2)^2+..+(a_n)^2)((b_1)^2+(b_2)^2+...+(b_n)^2)[/tex]

[tex]((a_1)^2+(a_2)^2+..+(a_n)^2)((b_1)^2+(b_2)^2+...+(b_n)^2)-(a_1b_1+a_2b_2+...+a_nb_n)^2 \geq 0[/tex]

I dont usnderstand how the below notation works as I cant follow from the above line to the line below , if someone can point me to some resources where I can know more about it :) ...

[itex]\displaystyle\sum_{i\not=j}^n ((a_i)^2(b_j)^2+(a_j)^2(b_i)^2-2a_ib_ja_jb_i ) [/itex]

Thanks

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# Cauchy -schwarz inequality help

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