Dot product is defined $\sum_{n=1}^\infty x_ny_n$. Let V consist of all sequences for which $\sum_{n=1}^\infty x_n^2$ converges.
Prove that this series converges absolutely.
I know I need to use the Cauchy-Schwarz inequality. But I don't see either how that helps or why?
Using the arithmetic geometric mean inequality, you get $$|x_ny_n|\leqslant \frac{x_n^2+y_n^2}2$$ so that $$2\sum_{n\geqslant 1}|x_ny_n|\leqslant {\sum_{n\geqslant 1}x_n^2+\sum_{n\geqslant 1}y_n^2}<+\infty$$
Using Cauchy-Schwarz you get $$\sum_{n\leqslant k}|x_ny_n|\leqslant \left( \sum_{n\leqslant k}x_n^2\cdot \sum_{n\leqslant k}y_n^2\right)^{1/2}$$ so that $$\sum_{n\geqslant 1}|x_ny_n|\leqslant \left( \sum_{n\geqslant 1}x_n^2\cdot \sum_{n\geqslant 1}y_n^2\right)^{1/2}<+\infty$$