MHB Proof of Knaster-Tarski Theorem

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The discussion centers on proving that if F is a monotone function and C is the union of sets invariant under F, then F(C) equals C. A participant questions the correctness of their proof, specifically regarding the relationship between elements of C and subsets X of A. They express uncertainty about why an element B in C implies the existence of a corresponding subset X such that X is a subset of F(X). The conversation highlights the need for clarity in establishing this connection within the proof. The thread ultimately seeks to validate the proof's logic and completeness.
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Let $F:P(A)->P(A$) be monotone and $C$ be the union of sets whose image is invariant under F. Prove $F(C)=C$

https://i.stack.imgur.com/3Wjdg.png
 
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What is your question?
 
Evgeny.Makarov said:
What is your question?

Hi, my question is my proof (in the image) correct?
 
Fermat said:
$C$ be the union of sets whose image is invariant under F
So, if I understand correctly, $$C=\bigcup_{X\subseteq A}F(F(X))=F(X)$$. But then it is not clear why $B\in C$ implies $B\in X$ for some $X\subseteq A$ such that $X\subseteq F(X)$.

Why don't you type the proof as text?
 

Thread 'A variant of the Monty Hall problem'

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