Every inner product space is a metric space. Show that every inner product space is a metric space.
To show this should I set the distance metric as $d(x,y) = <x-y,x-y>$, then show properties of being metric space such as d(x,y) = d(y,x) etc.?
If so the point I do not understand is why we set metric as $d(x,y) = <x-y,x-y>$ (this metric is mentioned in wolfram)

The inner product induces a norm via
$$||x||:=\sqrt{\langle x,x\rangle}$$
and then, the norm induces a metric via
$$d(x,y):=||x-y||$$
which is precisely what is mentioned in the comment above. In general, one has the relation $\text{Inner Product Spaces}\subset\text{Normed Vector Spaces}\subset\text{Metric Spaces}$.
