# ${x_{k}}\rightarrow x$ in $R^n$ if and only if ${x_{k}}\cdot y\rightarrow x\cdot y$ for all $y\in R^n$

Show that $${x_{k}}\rightarrow x$$ in $$R^n$$ if and only if $${x_{k}}\cdot y\rightarrow x\cdot y$$ for all $$y\in R^n$$

I think the following:

Given the $${x_{k}}\rightarrow x$$ this is equal to $$\lim\limits_{k\rightarrow \infty}x_{k}=x$$, then given a $$y\in R^n$$, we have to:
$$\begin{eqnarray*} {x_{k}}\rightarrow x &\Leftrightarrow& \lim\limits_{k\rightarrow \infty}x_{k}=x\\ &\Leftrightarrow& \lim\limits_{k\rightarrow \infty}x_{k} \cdot y = x \cdot y\\ &\Leftrightarrow& {x_{k}}\cdot y \rightarrow x \cdot y \end{eqnarray*}$$

Hint 1: Remember that a sequence in the metric space $$\mathbb{R}^n$$ (with the usual metric) converges to a point in the space if and only if the component sequences converge.
Hint 2: for every $$i= 1, \dots , n$$ make $$y = e_i$$, the vector whose k-entry is 1 whereas the other entries are 0.
Edit: By a component sequence of $$\{x_k\}$$ I mean the sequence whose $$k$$th term is the $$i$$th coordinate of the point $$x_k$$ from the original sequence (where $$i$$ is some fixed integer of the set $$\{1, \dots n\}$$). Thus, since you are dealing with a sequence of points on $$\mathbb{R}^n$$ (points described with n coordinates), you can construct $$n$$ of those sequences.