Limit Superior and Limit Inferior of a Sequence

We discuss the notion of the limit superior (limsup or upper limit) and limit inferior (liminf or limit inferior) of a bounded set of real numbers. For any sequence these limits will exist while the limit itself may not. The limit superior is limit of the subsequence which converges to the largest value and the limit inferior is the limit of the subsequence converging to the smallest value. We prove that these limits exist.

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