# Determine whether metric spaces with metrics of the form $d(x,y)=|f(x)-f(y)|$ are complete

How to decide if the metric spaces $((0,1)$, $d(x,y)=|x^2-y^2|)$ and $((-\frac{\pi}{2},\frac{\pi}{2})$, $d(x,y)=|\tan x-\tan y|)$ are complete or not?

For the first metric, I let any Cauchy sequence $(x_n)$ in $((0,1),d)$, then by definition I have for any small number $s>0$, there is $N>0$ such that for all $n,m>N$, $d(X_n,X_m) = |(X_n)^2-(X_m)^2| < s$. Then I have the sequence $((X_n)^2)$ being Cauchy in $((0,1)$, $d(x,y)=|x-y|)$. Then what?

• This is a must read. – Julien Apr 16 '13 at 18:45
• You have to figure out if such a Cauchy sequence would necessarily have a limit or not. – ferson2020 Apr 16 '13 at 18:54

• @Elvis Basically yes, although I would explain why $(x_n)$ diverges in different words, without mentioning $0$ explicitly. I have this feeling that whenever possible, properties of a metric space should be proved internally, i.e. using only points of that space. – Dan Shved Apr 17 '13 at 4:16
• Hint for the second metric space: $((-\pi/2,\pi/2),d)$ is isometric to $(\mathbb{R}, d_{\mathbb{R}})$ where $d_\mathbb{R}$ is the standard distance in $\mathbb{R}$. – Dan Shved Apr 17 '13 at 4:33