Banach Fixed-Point Theorem

Theorem
Let $\struct {M, d}$ be a complete metric space.

Let $f: M \to M$ be a contraction.

That is, there exists $q \in \hointr 0 1$ such that for all $x, y \in M$:


 * $\map d {\map f x, \map f y} \le q \, \map d {x, y}$

Then there exists a unique fixed point of $f$.

Uniqueness
Let $f$ have two fixed points $p_1, p_2 \in M$.

It is to be proved that $p_1 = p_2$.

Then from $\paren {1 - q} > 0$ and $\map d {p_1, p_2} \ge 0$:
 * $\map d {p_1, p_2} = 0$

From Metric Space Property $(\text M 4)$:
 * $p_1 = p_2$

Existence
The fixed point $p$ will be found from an arbitrary member $a_0$ of $M$ by iteration.

The plan is to obtain $\ds p = \lim_{n \mathop \to \infty} a_n$ with definition $a_{n + 1} = \map f {a_n}$.

The sequence of iterates converges in complete metric space $M$ because it is a Cauchy sequence in $M$, as is proved in the following.

Induction on $n$ applies to obtain the contractive estimate:


 * $\map d {a_{n + 1}, a_n} \le q^n \map d {a_1, a_0}$

Induction details $n = 1$:

Assume the contractive estimate for $n = k$.

Induction details for $n = k + 1$:

Next it is to be proved that $\sequence {a_n}$ is a Cauchy sequence in $M$ by showing that $\ds \lim_{m \mathop \to \infty} \map d {a_{n + m}, a_n} = 0$ for $n$ large.

We have that:
 * $\ds \lim_{n \mathop \to \infty} q^n = 0$

and:
 * $\dfrac {1 -q^m} {1 - q} \le \dfrac 1 {1 - q}$

Then $\sequence {\map d {a_{n + m}, a_n} }$ has limit zero at $m = \infty$ for large $n$.

Sequence $\sequence{a_n}$ is a Cauchy sequence convergent to some $p$ in $M$.

Then:

The has limit zero at $n = \infty$.

Then:
 * $\map d {\map f p, p} = 0$

Then by Metric Space Property $(\text M 4)$:


 * $\map f p = p$.