If a space is complete with respect to some metric is it complete with respect to any other metric?

Question

In proving if a metric space is complete the defined metric on it is considered. For example $\mathbb{R^n}$ is complete with respect to the 'standard' Euclidean metric. I was wondering if being complete is intrinsic to a space such that it is enough to show that it is complete with respect one metric so it will be complete with respect any other metric?

Otherwise, is there any counterexample?

Answer 1

Consider $\mathbb{R}^n-\{0\}$ it is not complete in respect to the standard Euclidean metric but is complete in respect of the metric defined by the riemmanian metric ${{\langle, \rangle}\over{\|x\|}} $.

Answer 2

$\mathbb{R}$ is complete with respect to the usual metric but is not complete with respect to the metric $$d(x,y)=\frac{|x-y|}{\sqrt{1+x^2}\sqrt{1+y^2}}$$. (Take the sequence $a_n=n$. It's Cauchy but doesn't have a limit)