Let $A=(0,\infty) $ be a given set on which we define a metric $d_1 (x,y)= |\log x -\log y| $. Prove that $d_1$ is complete.
MY TRY: _____
Consider the usual distance on A by $d_2 (x,y)=|x-y|$. Clearly $d_1 (x,y)= d_2 (f (x),f (y)) $ ,where $f (x)=\log x $. Thus $d_1$ and $d_2$ are isometric. $d_2$ is not complete and as isometry preserves completeness $d_1$ is also not complete. So I have disproved what I had to prove. I found this problem in a textbook, so I guess the problem is correct and somehow I have done something wrong! But how! Thanks for reading.
The error you are making is that $f(x)=\log(x)$ is a bijection from $A$ to $\mathbb R$. That means the isometry is an isometry between $(A,d_1)$ and ($\mathbb R,d_2)$, the latter being complete.