general topology – Open subset of a polish space

HERE is the theorem about being polish of open subsets of polish spaces. Let $X$ be a Polish space and the complete metric on $X$ is $d<1$ and the new one defined on any open set $Usubset X$ is as $d^{*}(x,y)=d(x,y)+ |frac{1}{d(x,Xsetminus U)}-frac{1}{d(y,Xsetminus U)}|$
clearly it is a metric on $U$ so it says that for each $i$, $frac{1}{d(x_{i},Xsetminus U)}$ is greater than $1$, so bounded away from zero, how is that $d(x, Xsetminus U)>0$ ($x$ is the limit of $x_{n}$ in $X$.