Final answer:
A monotonic sequence converges if and only if it is bounded. An increasing monotonic sequence that is bounded above converges to the least upper bound; similarly, a decreasing sequence that is bounded below converges to the greatest lower bound.
Step-by-step explanation:
The question is asking us to prove that a monotonic sequence converges if and only if it is bounded. A sequence is said to be monotonic if it is either entirely non-increasing or non-decreasing. To prove the statement, we can consider two separate cases for monotonic sequences: increasing and decreasing.
Case 1: Increasing Monotonic Sequence
If an increasing sequence {Sₙ} is bounded above, that is, there exists a real number M such that Sₙ ≤ M for all n, then by the Monotone Convergence Theorem, {Sₙ} will have a least upper bound, which is the limit to which the sequence converges. Conversely, if the sequence is convergent, i.e., it approaches a specific value L as n tends to infinity, then L serves as an upper bound, ensuring that the sequence is bounded above.
Case 2: Decreasing Monotonic Sequence
Similarly, if a decreasing sequence {Sₙ} is bounded below, there exists a real number N such that Sₙ ≥ N for all n, and by the same theorem, it will converge to the greatest lower bound. If the sequence converges to a limit L, then L acts as a lower bound, confirming that the sequence is bounded below.
In conclusion, regardless of whether the monotonic sequence is increasing or decreasing, it will converge if and only if it is bounded, which means it has either an upper bound or a lower bound, respectively.