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?  

Looks like you've plunged into epsilons, deltas and abstract reasoning from the very start. But it may be a good idea to simply check out one or two concrete sequences first, and see if you can come up with a counterexample to completeness right away.
Hint: if you have an open interval, weird things will usually happen near its endpoints. 
