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Under which conditions can we know that the limit of a pseudo convergent set of a valued field is unique?

Is there an example of a valued field in which any pseudo convergent set has a limit and such that this limit is unique?

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# uniqueness of a limit of a pseudo convergent set

Under which conditions can we know that the limit of a pseudo convergent set of a valued field is unique?

Is there an example of a valued field in which any pseudo convergent set has a limit and such that this limit is unique?