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I Know that all the norms are equivalent in R, so every distance deduced by a norm makes R complete. But what about the other distances ? A proof or a controexample would be appreciate

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    Are you asking if metrics on the reals exist which are not complete? If so then yes, tons and tons of them. – 2017-02-04
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    I'm not sure that this works but try with $d(y,x) =| \arctan y - \arctan x|$, if it doesn't there are many other examples. – 2017-02-04

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As I said in the comments, take $d(x,y) = |\arctan x - \arctan y|$. Check it is a distance. Take now $s_n = n, n \in \mathbb{N}$. Is it Cauchy? does it converge?

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Another example would be $d(x,y)=|e^{-x}-e^{-y}|$, and $x_n=n$ (easier to work with).

Then: $$d(x_m,x_n)=|e^{-m}-e^{-n}|\le e^{\min(m,n)}\to 0$$ as $m,n\to\infty$, so $x_n$ is Cauchy, and assuming $x_n\to x\in\mathbb R$ one gets $d(x_n,x)\to 0$, so: $$|e^{-m}-e^{-x}|\to|0-e^{-x}|=e^{-x}=0$$ which is of course impossible.