Completeness of two metrics in the same space

Question

Let $d, d'$ two metrics on the same space $X$, and suppose that $(X,d)$ is complete. I've read that if every ball $B'_r(x_0)=\left\{x\in X | d'(x_0,x)<r\right\}$ is contained in a ball $B_R(x_0)=\left\{x\in X | d(x_0,x)<R\right\}$, then $(X,d')$ is also complete. How does the completeness follow from the above property?

Answer 1

Suppose that $d$ is the discrete metric. Then it is complete of course. Moreover, for any metric $d'$, we have the property you described above (letting $R = 1.5$ for instance). But clearly there are non complete metrics. So what you say is not true.