(a) Suppose that {an} is a monotone sequence such that 1 ≤an ≤2 . Must the sequence converge? If

step 1 Okay, suppose I have a monotonic sequence that is bounded below and above. What does monotonic m

(a) Suppose that {an} is a monotone sequence such that 1 ≤an...

(a) Suppose that $\left\{a_{n}\right\}$ is a monotone sequence such that $1 \leq a_{n} \leq 2 .$ Must the sequence converge? If so, what can you say about the limit? (b) Suppose that $\left\{a_{n}\right\}$ is a monotone sequence such that $a_{n} \leq 2 .$ Must the sequence converge? If so, what can you say about the limit?

(a) Suppose that $\left\{a_{n}\right\}$ is a monotone sequence such that $1 \leq a_{n} \leq 2 .$ Must the sequence converge? If so, what can you say about the limit?
(b) Suppose that $\left\{a_{n}\right\}$ is a monotone sequence such that $a_{n} \leq 2 .$ Must the sequence converge? If so, what can you say about the limit?

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Key Concepts

Monotone Convergence Theorem

The Monotone Convergence Theorem states that every monotone sequence that is bounded (above if increasing, below if decreasing) converges. For an increasing sequence bounded above, the limit is the supremum, and for a decreasing sequence bounded below, it is the infimum.

Monotone Sequences

Monotone sequences are sequences that are either non-decreasing or non-increasing. Their predictable behavior makes it easier to analyze their convergence, as the sequence moves consistently in one direction without oscillation.

Bounded Sequences

A bounded sequence has its terms confined within fixed limits, meaning there exist real numbers that serve as an upper or lower bound for all terms. Boundedness is a critical property because it often prevents divergence, especially when combined with monotonicity.

Supremum and Infimum

The concepts of supremum (least upper bound) and infimum (greatest lower bound) are essential in determining the limits of monotone sequences. They provide the exact boundary—the limit—to which the sequence converges if the sequence is monotonic.

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