Is my proof for $\sum_{i=1}^{n}x_{i}y_{i}\leq \sqrt{\sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}}$ correct?

Is this part of my proof by induction correct ?

$\sum_{i=1}^{n}x_{i}y_{i}\leq \sqrt{\sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}}$

this is true when the true is that :

$\sum_{i=1}^{n}\left |x_{i}y_{i}\right |\leq \sqrt{\sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}}$

above inequality is true for $n=1$ and we assume that it's true for $n$.

For $n+1$ we get : $\sum_{i=1}^{n}\left |x_{i}y_{i}\right |+\left |x_{n+1}y_{n+1}\right |\leq \sqrt{\sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}+x_{n+1}^{2}\sum_{i=1}^{n}y_{i}^{2}+y_{n+1}^{2}\sum_{i=1}^{n}x_{i}^{2}+x_{n+1}^{2}y_{n+1}^{2}}$

using induction assumption we get :

$\sum_{i=1}^{n}\left |x_{i}y_{i}\right |+\left |x_{n+1}y_{n+1}\right |\leq \sqrt{\sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}}+\left |x_{n+1}y_{n+1}\right |\leq \sqrt{\sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}+x_{n+1}^{2}\sum_{i=1}^{n}y_{i}^{2}+y_{n+1}^{2}\sum_{i=1}^{n}x_{i}^{2}+x_{n+1}^{2}y_{n+1}^{2}}$

Is this correct ? Someone told me that I've used induction in wrong manner.

So many extra parentheses in these equation. Why write $(x_i)^2$ rather than $x_i^2$? –  Thomas Andrews May 7 '12 at 15:02