$d(x,y)=\|x-y\|$, $x,y \in \mathbb{R}^n$. 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?
$d(x,y)=\|x-y\|$, $x,y \in \mathbb{R}^n$. 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?
$\forall \, x,y,x_0,y_0 \in \mathbb{R^n},$ we have
$ |d(x,y)-d(x_0,y_0)|=|\, \|x-y\|-\|x_0-y_0 \| \,| \leq |\, ||(x-y)-(x_0-y_0)|| \,| $
$ = |\, ||(x-x_0)-(y-y_0)|| \,| \leq \| x-x_0\| + \| y-y_0\|< \delta_1 + \delta_2 =\epsilon,$
where $\delta_1=\delta_2=\frac{\epsilon}{2}.$