Theorem 1 (Existence of a fixed point)

Theorem 1: If [Maple Math] is continuous on the interval [Maple Math] and [Maple Math] is an element of [Maple Math] for all values of [Maple Math] in the interval [Maple Math] , then g has a fixed point in [Maple Math] .

Proof:

if [Maple Math] or [Maple Math] then a fixed point exists.

Otherwise, we must have that [Maple Math] and [Maple Math] .

Define [Maple Math] . Then [Maple Math] and [Maple Math] .

Therefore [Maple Math] and [Maple Math] . Since [Maple Math] is continuous, there must be a point in [Maple Math] where [Maple Math] .

Calling this point [Maple Math] , we have [Maple Math] or [Maple Math] . therefore, there exists a fixed point of [Maple Math] , i.e. [Maple Math] .