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$d(x,y)=\|x-y\|$, $x,y \in \mathbb{R}^n$. How can I show $\|\cdot\|$ is continuous jointly in $x$ and $y$?

I have written the following:

$\|x-y\|, $d(\|x-y\|,\|x_0-y_0\|)<\epsilon$ isn't that true? How should I continue?

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    It could be helpful to explain the line that you have written to somebody, e.g, to MSE.2012-12-25

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