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$A$ is a poset where every chain has an upper bound in $A$ and $a$ is any element in $A$. let $$C_a=\{x∈A∣a≤x\}$$ Can some one help me how to show $(C_a,\le )$ is a poset and how it satisfies hypothesis of Zorn's Lemma?

To show $(C_a, \le)$ is a poset I know I should show it is reflixive, antisymmetric and transitive. But I dont know how to even start it. Can any help me?

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Are you ordering the $C_a$ or are you restricting the domain to a particular $C_a$? –  vadim123 May 9 '13 at 0:35
    
I am restricting domsin to a particular Ca. –  Ahmed May 9 '13 at 1:03
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1 Answer

The poset properties are inherited directly from $A$. What's more difficult is that every chain in $C_a$ has an upper bound in $C_a$. It has an upper bound in $A$, since $A$ has that property, but that upper bound needs to be in $C_a$.

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