Consider the space $X=[0,1]$ Then
A) $\left\{\frac{1}{n}\right\}$ converges to $\frac{1}{2}$ in some metric space $(\mathrm{X}, \mathrm{d})$
B) $\left\{\frac{1}{n}\right\}$ converges to 1 in some metric space $(X, d)$
C) $\left\{\frac{1}{n}\right\}$ is not Cauchy in some metric space $(\mathrm{X}, \mathrm{d})$
D) All of these
For the first one For any one-to-one function $f d(x, y)=|f(x)-f(y)|$ defines a metric. I can choose a function $f$ such that $f\left(\frac{1}{n}\right) \rightarrow f\left(\frac{1}{2}\right)$ as $n \rightarrow \infty$. Hence first option correct.
What about the other options ?
Thanks in advance