Final answer:
` Yes, for a convergent sequence.` The limit superior (lim sup) and limit inferior (lim inf) always exist for a convergent sequence.
The correct answer is B
Step-by-step explanation:
For a convergent sequence, the lim sup and lim inf always exist and are equal to the limit of the sequence. However, for a non-convergent sequence, the lim sup and lim inf may not exist.
1. A sequence is convergent if it approaches a specific value as the index of the sequence increases. In other words, the terms of the sequence get arbitrarily close to a fixed number.
2. The limit superior (lim sup) of a sequence is the largest limit point that the sequence can approach. It represents the supremum (or maximum) of the set of all possible subsequential limits of the sequence.
3. The limit inferior (lim inf) of a sequence is the smallest limit point that the sequence can approach. It represents the infimum (or minimum) of the set of all possible subsequential limits of the sequence.
4. For a convergent sequence, the limit superior and limit inferior are equal to the limit of the sequence. This means that both lim sup and lim inf exist and are equal to the same value.
5. However, for a non-convergent sequence (divergent sequence), the lim sup and lim inf may not exist. In this case, the sequence may have multiple subsequential limits, or the limits may be infinite.
Therefore, the correct answer is B) Yes, for a convergent sequence, as the lim sup and lim inf always exist in this scenario.