A contraction is a function such that for all and some .

For uniqueness, assume (for a contradiction) and . What can you say about ?

As for existence, define a sequence such that . Try to find a formula in terms of and for and use this to show that is Cauchy (and therefore converges to some point ). Then use the continuity of to show that .

What you are proving is known as theContraction Mapping Principleor alternatively, theBanach Fixed Point Theorem.