Final answer:
If a sequence is both monotone and bounded, it is convergent, meaning it approaches a finite limit. The Monotone Convergence Theorem applies to such sequences, ensuring they do not oscillate and approach a specific real number.
Step-by-step explanation:
From a mathematical standpoint, if a sequence is both monotone and bounded, you can deduce that it is convergent. This means that as the sequence progresses with each term, it approaches some finite limit. A monotone sequence only moves in one direction - either always increasing or always decreasing.
The boundedness refers to the sequence having an upper or lower limit that it will not exceed. Together, these properties ensure that the sequence will not oscillate wildly and is approaching a specific real number. Such sequences confirm to the Monotone Convergence Theorem.
For example, consider a sequence where each term is the inverse of its term number ({1/n} for the n-th term). This sequence is decreasing (therefore monotone) and bounded below by 0. As n increases, the terms get closer and closer to 0 without ever reaching it, demonstrating both monotonicity and boundedness leading to convergence at 0.