How do we prove the continuity of the inner product? Let $\lim x_{k}=a$, $\lim y_{k}=b$ in $\mathbb{R^n}$ then it must be proved that $\lim \langle x_{k}, y_{k}\rangle = \langle a, b\rangle$, where $\langle a, b \rangle$ is the inner product in $\mathbb{R^n}$
The idea I have is that $ \langle x_{k}, y_{k} \rangle$ $=$ $\sum_{i=1}^{n} x_{i}^k y_{i}^k$ then $\lim \langle x_{k},y_{k}\rangle = \lim \sum_{i=1}^{n} x_{i}^k y_{i}^k$ $=$ $\lim x_{1}^ky_{1}^k+...+\lim x_{n}^ky_{n}^k$
That would be fine and being like that how I continue?

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*Show that $f(x,y)=\|x+y\|^2$ and $g(x,y)=\|x-y\|^2$ are continuous from $(\Bbb R^n)^2$ to $\Bbb R.$


*The inner product $<x,y>=\frac {f(x,y)-g(x,y)}{4}.$
