Suppose you know that {an} is a decreasing sequence and all its terms lie between the numbers 5

step 1 For this problem, you're given that A .N is decreasing. And here, you see that it's between 5 an

Suppose you know that {an} is a decreasing sequence and all...

Suppose you know that $\left\{a_n\right\}$ is a decreasing sequence and all its terms lie between the numbers 5 and 8 . Explain why the sequence has a limit. What can you say about the value of the limit?

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Key Concepts

Bounded Sequences

A bounded sequence is one that has all its terms lying within some fixed interval. In real analysis, a sequence that is bounded above and below does not diverge to infinity or negative infinity, which is a necessary condition for convergence when combined with other properties.

Monotone Sequences

A monotone sequence is either entirely non-increasing or non-decreasing. In a decreasing sequence, each term is less than or equal to the previous term. This property simplifies the analysis of its behavior because the changes in the sequence are in a single direction.

Monotone Convergence Theorem

The monotone convergence theorem states that every monotone sequence that is bounded converges to a limit. Since the given sequence is both bounded and decreasing, this theorem guarantees that the sequence has a limit.

Limit of a Convergent Sequence

When a sequence converges, its limit is a number that the sequence's terms get arbitrarily close to as the index increases. Since the sequence's terms are confined between 5 and 8 and are decreasing, the limit must lie within this interval, being at least 5 and at most 8.

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