Existence of unique fixed point in compact Metric space

Question

Let $(X,d)$ be compact. Show: for a map $f$ that when $\forall x, y \in X$ with $x\neq y$

$d(f(x),f(y))<d(x,y)$ is fulfilled.

Then $f$ has a unique fixed point.

https://math.stackexchange.com/questions/118536/prove-the-map-has-a-fixed-point?rq=1 — Tsemo Aristide, May 22 '18 at 08:24

score 1 · Answer 1 · answered May 22 '18 at 08:25

1

Assume there are two fixed points, let $x$ and $y$.

As $f(x)=x$ and $f(y)=y$, then $$d(f(x),f(y))=d(x,y)<d(x,y),$$ a contradiction.

answered May 22 '18 at 08:25

I don't see how compactness matters. – May 22 '18 at 08:28
The statement $f$ has a unique fixed point means $f$ has a fixed point and it does not have more than one fixed point. Where did you prove the existence of a fixed point? – Kavi Rama Murthy May 22 '18 at 08:58
@KaviRamaMurthy: the title led me to think that only uniqueness was to be established. – May 22 '18 at 09:02
Even the title talks about both existence and uniqueness. – Kavi Rama Murthy May 22 '18 at 09:40
Extremely sorry. I didn't know the the title was changed. – Kavi Rama Murthy May 22 '18 at 11:39

1 Answers1