Let $(X,\rho)$ and $(X,\sigma)$ be metric spaces. If $X$ is $\rho$-complete is $X$ $\sigma$-complete? Justify your answer.
A little bit of reference showed that this is not necessarily the case. So I have to give a counter example of a space which is $\rho$-complete but not $\sigma$-complete.
Any hints/ideas? Thanks in advance.