Let $X=[1,\infty)$, and $d(x,y)=|\frac{1}{x} -\frac{1}{y}|$.
Is this set $X$ complete? Is it compact?
I have difficulty understanding what completeness really is. I know the defintion that within the complete set every Cauchy sequence converges. I am using Rudin's book, there it is specified as an example that the metric space of rational numbers with the distance function $d(x,y)=|x-y|$ is not complete. I do not understand this. If someone could address these questions I have, I would be grateful.
The distance function is always smaller than one. I assume the set is not compact because it is not bounded above? Or is it bounded by $1$?
Thanks in advance!
Best Answer
To show that a set $X$ is complete, you have to take an arbitrary Cauchy sequence $\{x_n\}$ with elements in the set and do $3$ things:
1) search for a hypothetical $x$ which you expect to be the limit of $x_n$
2) check that this $x$ is indeed an element of the set $X$
3) prove that $d(x_n,x)\to 0$
For compactness in a metric space, it is enough to show that each sequence $\{x_n\}\subset X$ has a convergent subsequence with a limit again in the set $X$.