I have found myself in a tiny problem. If $(X,d)$ is a metric space and $A\subset X $ not empty, then $(A,d´)$ with $d´=d_{AxA}$ is a metric space. Where $d_{AxA}$ is the metric $d$ restricted to subset A.
It´s pretty intuitive that if get restricted the metric space we should get another one but I got some trouble trying to prove it. Is it enough to say that $d´(x,y)=d_{AxA}(x,y)=0$ if only if $x=y$ with $x,y\in A$ ass $d$ is already a metric?
If that so then the triangular equalty is easy to prove it. Then We got the metric space. Is it right? or it is much harder?