Q:

extension of cauchy schwarz inequality

I Want to prove $(a_1^2+a_2^2)(b_1^2+b_2^2)(c_1^2+c_2^2)>_=(a_1b_1c_1+a_2b_2c_2)^2$ 
not quite sure how and not only upto 3 terms but to $n$ terms, I am not even sure if this inequality is true or not but I need this result to solve the following problem 

A:

RHS $\leq (a_1^{2}+a_2^{2})(b_1^{2}c_1^{2}+b_2^{2}c_2^{2})$ by Cauchy-Schwarz inequality. But $b_1^{2}c_1^{2}+b_2^{2}c_2^{2} \leq (b_1^{2}+b_2^{2})(c_1^{2}+c_2^{2})$ so we are done. 

