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of .
* Here, is not required to lie in the interior of . Indeed, if is an endpoint of , then the above limits are left- or right-hand limits.
* A similar statement holds for infinite intervals: for example, if , then the conclusion holds, taking the limits as .This theorem is also valid for sequences. Let be two sequences converging to , and a sequence. If we have , then also converges to .

Proof

According to the above hypotheses we have, taking the limit inferior

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