Let $(X,d)$ be any metric space and let $p\in X$ and let $d_p$ be the corresponding french railroad metric. That is, $d_p(x,y)=d(x,p)+d(p,y)$.
I'm supposed to show that $(X,d_p)$ is necessarily a complete metric space. So any Cauchy sequence in $(X,d_p)$ had better converge to some $x\in X$.
Can someone help me go about showing that there cannot be a Cauchy sequence whose limit is not in $X$?