The Banach fixed point theorem:

Let $(X,d)$ be a complete metric space and $$f: X \to X$$ be a contractive map then there is a unique fixed point. 

I am trying to prove the uniqueness of a fixed point.

This is the proof: 
Assume that $x\_1,x\_2$ are two distinct fixed points.
Then $$0<d(x\_1,x\_2)=d(f(x\_1), f(x\_2)) \leq c.d(x\_1,x\_2) < d(x\_1,x\_2)$$ 

I understand that a contractive map satisfies $d(f(x),f(y)) \leq c.d(x,y)$
where $x,y \in X$ and constant $c \in (0,1)$.

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> In the proof how can we say that $c\cdot d(x\_1,x\_2) < d(x\_1,x\_2)$?
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The proof finishes of by saying 

So $d(x\_1,x\_2)=0$ and so $x\_1=x\_2$.

How can we deduce this?