Prove the Uniqueness of a Contraction's Fixed Point on [a,b]

[nickolas2730]

Homework Statement

A function f :[a, b] → [a, b] is said to be a contraction on [a, b] if there exists a constant k. (0, 1) such that |f(y) - f(x)| <k|y - x| for all x, y in [a, b]. Let f be a contraction. Show that f is uniformly continuous on [a, b].

Let f : [a, b] → [a, b] be a contraction. Since it is a continuous
function by one of the previous exercises f has at least one fixed point. Prove that the fixed
point of a contraction is unique.

thank you so much

Homework Equations

The Attempt at a Solution

i tried the first question as following:

Let $$\epsilon$$>0 and set $$\delta$$=$$\epsilon$$/k,│y-x│< $$\delta$$ , which implies
│f(x)-f(y)│<k*│y-x│<k*$$\delta$$=]=$$\epsilon$$

but i have no idea with the second question, since i couldn't even find the word contration in my entire textbook

 

[Mark44]

Homework Statement

A function f :[a, b] → [a, b] is said to be a contraction on [a, b] if there exists a constant k. (0, 1) such that |f(y) - f(x)| <k|y - x| for all x, y in [a, b]. Let f be a contraction. Show that f is uniformly continuous on [a, b].

Let f : [a, b] → [a, b] be a contraction. Since it is a continuous
function by one of the previous exercises f has at least one fixed point. Prove that the fixed
point of a contraction is unique.

thank you so much

Homework Equations

The Attempt at a Solution

As you are a new member (welcome!) you probably haven't had a chance to see the Rules here. Click Rules in the menu bar to see them. Here's an except from the rules, in the section on Homework Help.

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[nickolas2730]

oh..thank you so much..
so, do i need to type out what i have done for the question now??
can i just type it here instead of the homework template?
or do i need to edit my post and type the work in "The attempt at a solution"?

 

[Mark44]

Yes, put in what you've done. I don't think you need to start all over again, but keep the template in mind when you post another problem.