Q. The metric space M consists of all ordered pairs x =$(x',x'')$ of real numbers with metric $\rho=max(|x'-y'|,|x''-y''|)$. Prove that M is complete.
My approach: We know that a space is complete if every Cauchy sequence in M has a limit in M. My doubt is how do I show that any arbitrary Cauchy sequence in M is converging to the limit which lies in M?