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An informal argument for Zorn's lemma can be given as follows: Assume that S is closed under unions of chains.

Assume that S is equicontinuous.

Suppose that S is hemicompact.

Suppose that S is continuous.

Suppose that S is nonempty.

Assume that S is ρ-continuous.

It is easy to see that S is nondecreasing.

end{aligned} We see that S is contractive.

Assume that S is equicontinuous at 0 ∈ X.

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