Let $C_b(X,R)$ be the set of all bounded continuous functions on $X$ with supremum norm. Is $l(X)$ dense in $C_b(X,R)$ with respect to the map $l:X \to C_b(X,R)$ defined by $l(x)=f_x$ where $f_x(t)=d(x,t)-d(x_0,t)$ where $x_0$ is a fixed point in $X$.
This is a step proving every metric space has a completion from Volker Runde's "A taste of topology". I checked that the map $l$ is an isometry and $C_b(X,R)$ is complete. The book also have not shown this. It is not obvious to me. Thanks in advance for your help.