Banach fixed-point theorem

Let F be a non-empty closed subset of a complete metric space (X, d). Let T : X -> X be a contraction operator on F, , i.e: There is a constant q ε (0, 1) such that

d(Tx, Ty) < q • d(x, y)

for all x, y ε X. Then:

1. For any x₀ ε F rhe sequence defined by x_n+1 = Tx_n, n ε {1, 2, 3, ...} remains within F

2. There exists an x* ε F such that:

lim_{n -> INFTY} x_n = x*

3. x* is a fixed point of T i.e. Tx* = x*

4. x* is the only fixed point of T in F

5.

                 qn
   d(x*, xn) < ----- d(x1,x0)
                1-q