Thread: Proof relative to metric spaces.

1. Proof relative to metric spaces.

I'm trying to do a proof on some metric spaces, I think I've almost completed it but I am stuck on the last part of it.

The two metric spaces I am working with are $(R^+,d_1)$ $(R^+,d_2)$ where $d_1(x,y)=|x-y|$ and $d_2(x,y)=|ln(y/x)|$

So far I can show that any Cauchy sequence in d2 is also a cauchy sequence in d1, in short because if $|ln(y/x)|$ converges to zero then (y/x) has to converge to one, which means that the difference between x and y has to converge to zero.

I can similarly show that whenever a sequence converges to some limit $l\in R^+$ (notice it cant be zero) in d1, then it converges to the same limit in d2.

Now the only thing that I need to prove next is that if a sequence is Cauchy in both metrics, it cannot converge to zero in d1, or equivalently if a sequence converges to zero in d1, it will not be Cauchy in d2.

I don't really know how to approach this; I see that if x and y are really close to zero, ln(x) and ln(y) will be very large, but I can't manage to turn this into something useful...

2. Re: Proof relative to metric spaces.

Notice that $\bigl|\ln(y/x)\bigr| = \bigl|\ln y -\ln x\bigr|$. It follows that if the sequence $(x_n)$ is Cauchy for d2, then the sequence $(\ln(x_n))$ is Cauchy for the usual metric d1 and therefore converges to a finite limit. So $x_n$ must be bounded away from 0.

3. Re: Proof relative to metric spaces.

Eureka, thank you very much.