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)

That is wrong. It should be $d(x,y)=\sqrt{\langle x-y,x-y\rangle}$ because the map $x\mapsto\sqrt{\langle x,x\rangle}$ is a norm. And, whenever you have a norm $\lVert\cdot\rVert$, the map $(x,y)\mapsto\lVert x-y\rVert$ is a distance.
