*X*be a complete metric space, and let

*f: X->X*be a contracting function (we shall be using the definitions and symbols of my writeup in that node, so you might want to read it). Our aim is to show that

*f*has a

*unique*fixed point. The proof gives a taste of constructive methods in complete metric spaces; non-constructive methods also abound, but have a vastly different character. In particular, we shall give a method which converges to the fixed point of

*f*; this method is in fact used, e.g. to draw fractals or solve differential equations.

Uniqueness is easy: suppose *x,y* are 2 fixed points of *f*. Then

withd(x,y) =d(f(x),f(y)) <=c d(x,y)

*c*<1, so

*d*(

*x,y*)=0 and

*x=y*--

*f*has at most one fixed point.

For existence, first note that any contracting function must be continuous. Pick *any* *x*_{0} in *X*, and define *x*_{n}=*f*(*x*_{n-1}).
If we can prove that this sequence converges, by applying *f* and using its continuity we see that

f(limso the limit is a (the) fixed point ofx_{n}) = limf(x_{n}) = limx_{n+1}= limx_{n},

*f*

Write *d*=*d*(*x*_{0},*x*_{1}).
Then utilising the definition of a contracting function *n* times, *d*(*x*_{n+1},*x*_{n}) <= *d c*^{n}.
But this means that if *m*>*n* then
*d*(*x _{m},x_{n}*) <

*d c*

^{n}/ (1-

*c*), so our sequence is a Cauchy sequence. Thus it has a limit, which we saw is the fixed point of

*f*.

Note that the above is a procedure for converging to the fixed point! Furthermore, we see that the rate of convergence is respectable -- a constant number of digits for each iteration.