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I am reading the book by Burago and Ivanov "A course in metric geometry". I tried to do some problems but have some difficulties. For example, page 66 exercise 3.1.26:

Let $(X, d)$ be a metric space and set $d_{\epsilon}$ to be the maximum metric w.r.t. the covering of $X$ by all balls of radius $\epsilon$ equipped with the restrictions of d. Let $d_{0}(x, y) = sup_{\epsilon > 0}( d_{\epsilon}(x, y))$. Prove that if $d$ is complete, then $d_0$ is the intrinsic metric induced by d.

The problem states that we have to prove that $d_{0}=d_{L}$ where $d_{L}$ is the metric induced by the length structure on the metric space $(X,d)$. I have no difficulties proving that $d_{L}\geq d_{0}$. But what about the other inequality, i.e. $d_{0} \geq d_{L}$. Here I have no idea. Can anyone show me how to use completness to prove this last inequality? Thanks to all of you.

Cheers Mark

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so far I have that if $d$ is complete then $d_{L}$ is also complete and by $d_{L} \geq d_{0}$ also $d_{0}$ is complete. But how to go on ? –  Mark May 28 '13 at 9:13
    
here is the link for the book: " math.psu.edu/petrunin/papers/alexandrov/bbi.pdf " –  Mark May 28 '13 at 9:16
    
does anyone have an idea? –  Mark May 28 '13 at 10:01
    
yes, I also thaught that one could use corollary 2.4.17 ... but how should one apply it? maybe by contradiction? –  Mark May 28 '13 at 12:50
    
or by using the completness of $(X,d_{L})$ ? –  Mark May 28 '13 at 12:52

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