Contraction Mappings without a fixed point

Show that the following mapping $\displaystyle f:X \rightarrow X$ do not have a fixed point and explain why the Contraction Mapping Principle is not contradicted.

Problem: Let X=R and f(x) = x+1 for all x in X.

My Solution:

f do not have a fixed point because x does not equal to x+1 for any real number x.

d(f(x),f(y)) = |(x+1) - (y+1)| = |x-y|,

and d(x,y) = |x-y|.

So d(f(x),f(y)) = d(x,y), now I know I have to find a constant c with 0 <= C < 1 for the Contraction Mapping Principle to hold. But I can't find that c.

Please help.