12 questions linked to/from Prove the map has a fixed point
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### Fixed point of continuous function on compact metric space [duplicate]

Possible Duplicate: Prove the map has a fixed point Let $X$ be a compact metric space and $f:X\rightarrow X$ be a continuous function such that $d(f(x),f(y))<d(x,y)$ for every $x,y\in X$ such ...
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### Existence of unique fixed point in compact Metric space [duplicate]

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.
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### A problem with a compact metric space and $f \colon X \to X$ a uniformly continuous function [duplicate]

Let (X,d) be a compact metric space and $f \colon X \to X$ a uniformly continuous function so that $d( f(x),(f(y)) < d(x,y) , \forall x,y \in X , x \neq y$ . Prove that there exists x' in X so ...
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### A subset of a compact set is compact?

Claim:Let $S\subset T\subset X$ where $X$ is a metric space. If $T$ is compact in $X$ then $S$ is also compact in $X$. Proof:Given that $T$ is compact in $X$ then any open cover of T, there is a ...
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### Prove there is no contraction mapping from compact metric space onto itself

This question is from Foundations of mathematical analysis by Richard Johnsonbaugh The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a ...
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### How can I prove $x_{n+1} = e^{-x_n}$ is convergent?

I'm doing a practice problem which asks to prove that the sequence defined by $x_{n+1} = e^{-x_n}$ is convergent (or rather "study the convergence of $(x_n)$"). So I'd like to try and find some ...
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### Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
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### Infimum of Distance in Compact Space

If $X$ is a compact metric space, $A: X\to X$, is it true that if $a = \inf d(x,Ax),\space x \in X$, then there exists $y \in X$ such that $d(y,Ay) = \inf d(x,Ax)$? If so, why?
### If $X$ is a compact space and $A:X\rightarrow X$ is such that $\rho(Ax,Ay)<\rho(x,y)$, $x\neq y, \Rightarrow A$ have only one fix point on X.
I have problems with this demostration can anybody help me please? If $X$ is a compact space and $A:X\rightarrow X$ is such that $\rho(Ax,Ay)<\rho(x,y)$, $x\neq y, \Rightarrow A$ have only one ...