Proving the Cauchy-Schwarz inequality by induction I ran across this problem in some old notes, and I frustratingly can't figure out how to do it 
Let $a_i$ and $b_i$ be sequences of natural numbers, use induction to show
$\sum_{i=1}^n (a_ib_i)^{1/2} \le (\sum_{i=1}^n a_i)^{1/2}(\sum_{i=1}^n b_i)^{1/2} $
Obviously this is trivial to show for n=1. I can't make much progress on n+1. I've tried various tactics, squaring both sides etc. 
Any hint or help would be appreciated. Thanks. 

Hint: Another way would be to rewrite using the substitutions $x_i^2=a_ib_i$ and $y_i^2=a_i^2$ in the form 
$$CS(n): \quad \frac {\left( \sum_{i=1}^n x_i \right)^2}{\sum_{i=1}^n y_i} \le \sum_{i=1}^n \frac{x_i^2}{y_i}$$
This form is particularly amenable to induction, after you prove the base case of $n=2$. 
