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.
I'm adding link provided by https://math.stackexchange.com/users/18986/david-mitra http://ajmaa.org/RGMIA/papers/v12e/Cauchy-Schwarzinequality.pdf