# Verification of completeness of $d_1 (x,y) = |\log x -\log y|$

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.