Compact subsets of metric space with French railway metric Let $A=\{0,1,2,...\}$ with $f$ the French railway metric that has centre $0$ and $f(a,0)=1$ for all $a\in A$ with $a\neq0$.

  
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*How do I show that the metric space $(A,d)$ is complete?  
  
*How do I show that $A$ is bounded, but not totally bounded?  
  
*What is an example of a sequence in $A$ without convergent subsequence?  
  
*What is an example of an open cover for $A$ without a finite subcover?  
  
*How do I find all the compact subsets of $A$?
  

What I know: For the French railway metric we know that for $a,b\in A$ with $a\neq b$ we have $$f(a,b)=f(a,0)+f(0,b)=1+1=2$$ (provided that neither $a$ nor $b$ are $0$, otherwise $f(a,b)=1$).  


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*Being complete means that all Cauchy sequences in $A$ converge. I think the trick to proving this lies in the above described French metric, but I don't see it at the moment.  

*Being totally bounded means that there are finite $a_1,...,a_n$ in $A$ such that $A=\bigcup_{i=1}^nB_\epsilon(a_i)$. It seems intuitive that this is not the case, but how do I prove this exactly?  

*Sadly, I have no idea how to handle this one.  

*An open cover is a collection $X$ of open subsets of $A$ such that $A\subset\bigcup_{U\in X}U$. Then we need to find one such that no finite subcolection of $X$ is an open cover for $A$.  

*I was thinking we could take all the finite subsets of $A$; since being compact means that for all open covers of such a subset, there are finitely many elements in that open cover such that these elements are an open cover. But I am not at all sure of this, or even wheteher these would be all the compact subsets.


HINT: For $(1)$, the key is to figure out which sequences are Cauchy. If $a,b\in A$, and $f(a,b)<1$, what can you say about $a$ and $b$?
The answer to the following question is the key to answering every other part of the problem:


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*If $a\in A$ and $0\le\epsilon\le 1$, what is the set $B_\epsilon(a)$?

