(Solved): 2. Let V be a vector space over R equipped with a positive definite symmetric bilinear form :VV ...

2. Let $$V$$ be a vector space over $$\mathbb{R}$$ equipped with a positive definite symmetric bilinear form $$?:V\times V?\mathbb{R},$$ and let $$?$$. II be the associated norm on $$V$$. We say that a sequence $${\left\{x_n\right\}}_{n=1}^{?}$$ in $$V$$ converges to $$x?V$$ and write $$x_n?x$$ if $$?x?x_n??0$$ as $$n??$$. Suppose now that we have sequences $$x_n?x$$ and $$y_n?y$$ in $$V$$. Show that $$?\left(x_n,y_n\right)??(x,y)$$. (Hint. Observe that $$?(x,y)??\left(x_n,y_n\right)=?\left(x?x_n,y\right)+?\left(x_n,y?y_n\right)$$, and then use the Cauchy-Schwarz inequality.)