Prove that metric space is complete

Question

I have metric space: $$ X = <[0,+\infty), \rho>, \rho(x,y) = |ln(1+x) - ln(1+y)|$$ I know it is complete, but I don't know how to prove it. How can I prove that fact?

Answer 1

If $\{x_k\}$ is Cauchy in the metric $\rho$, then $\{\ln(1 + x_k)\}$ is Cauchy in the standard metric. Thus there exists a real number $y$ with the property that $\ln(1 + x_k) \to y$. Show there exists $x \in [0,\infty)$ with $\ln (1+x) = y$ and $\rho(x_k,x) \to 0$.