Is this a complete metric space?

Question

Given the space $\mathbb{R}_{++} = (0,\infty)$, and the metric $d(x,y) = |ln(\frac{x}{y}|$ , is $(\mathbb{R}_{++}, d)$ a complete metric space? What is a necessary and sufficient condition for a subset $E$ to be bounded? What is a necessary and sufficient condition for $E$ to be compact?

I would think that the answer to the first question is "no", because one could create a Cauchy sequence in this space that converges to $0$, which is not in the space itself.

For the second question, I would think that all that is necessary would be for the subset $E$ to have a supremum to be bounded. No specific infimum would be needed since any subset in this space is already bounded below by $0$.

The last questions is hardest for me. I would guess that being closed and bounded would be enough, by definition. This question was meant to help teach about suprema and infima, so maybe another way to answer this part would be to say: in this space, a subset will be compact if it's supremum is a maximum and it's infimum is a minimum...?

Thank you for your input!

Answer 1

Consider $f:(0,+\infty)\to R$ defined by $f(x)=\ln x$. Then $f$ is bijective and $|f(x)-f(y)|=|\ln x-\ln y|=|\ln {x\over y}|=d(x,y)$. This implies that $f$ is an isometry between $((0,+\infty),d)$ and $(R,| |)$ since $(R, | |)$ is a complete metric space, so is $((0,+\infty),d)$.