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Sequential criterion for limits and axiom of choice

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This may be an absurd question but it doesn't let me sleep. Why doesn't the usual proof of the sequential criterion of (real) limits requiere the axiom of (countable, in that case) choice? A professor told me that it follows by the axiom schema of replacement, but that doesn't convince me since it requires uniqueness.

If possible, please give an axiom-based construction of the sequence of the proof.


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