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Answer :
Final answer:
A bounded sequence (an) converges if and only if its limit superior and limit inferior are equal, indicating the sequence approaches a single value. The correct option is B, which states 'Lim sup an = Lim inf an'.
Explanation:
The question is concerned with the convergence of a bounded sequence (an). A bounded sequence is one in which all its terms lie within some fixed real number bounds. For a bounded sequence to converge, the limit superior (lim sup an) and the limit inferior (lim inf an) must be equal, which means that the sequence approaches a single value from above and below. If lim sup an and lim inf an are equal to each other, the sequence converges, and the common value is the limit of the sequence (lim an).
The correct answer is B. Lim sup an = Lim inf an, as it is only under this condition that a bounded sequence can be said to have a converging limit, denoted as Lim an.
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