Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$
Solution 1:
HINT: If $x_n\to x_0$ in the standard metric on $(0,\infty)$, then $\ln x_n \to \ln x_0$.
Solution 2:
Let $(x_n) $ be a Cauchy sequence then $d(x_n, x_m)<\epsilon$ so $\mid \ln(x_n)-\ln(x_m)\mid <\epsilon.$ But $$\mid \ln(x_n)-\ln(x_m)\mid=\mid \ln\frac{x_n}{x_m}\mid<\epsilon$$ So $\frac{x_n}{x_m} \rightarrow 1$ then subsequencec $(x_n)$ and $(x_m) $ have same limit and the sequence is convergent.
Solution 3:
Hint If $x_n$ is Cauchy, prove that $e^{x_n}$ is Cauchy with the usual metric of $\mathbb R$. If $y$ is the limit of $e^{x_n}$ then.....