Final Answer:
The Monotone Convergence Theorem states that if a sequence (a_n) is monotonically increasing and bounded above, then it converges to a limit. Similarly, if a sequence is monotonically decreasing and bounded below, then it converges to a limit.
Step-by-step explanation:
The Monotone Convergence Theorem is a fundamental result in real analysis that establishes the convergence of certain monotonic sequences. Consider a monotonically increasing sequence (a_n) that is bounded above. This means that for all n, a_n ≤ a_(n+1), and there exists an upper bound M such that a_n ≤ M for all n. The goal is to show that this sequence converges to a limit.
To prove this, we can utilize the completeness property of the real numbers. Since the sequence is bounded above, it has a least upper bound, denoted as L. Now, let's show that L is the limit of the sequence. Given any ε > 0, there exists an N such that for all n ≥ N, |a_n - L| < ε. This demonstrates that as n approaches infinity, the terms of the sequence get arbitrarily close to L, establishing the convergence of the sequence.
The proof for a monotonically decreasing sequence bounded below follows a similar logic. In this case, the sequence has a greatest lower bound, denoted as l, and the same argument can be applied to show convergence. The Monotone Convergence Theorem provides a powerful tool for analyzing the behavior of monotonic sequences, showcasing the relationship between monotonicity, boundedness, and convergence in the context of real analysis.
Complete Question:
Prove the Monotone Convergence Theorem: If a sequence (a_n) is monotonically increasing and bounded above, then it converges to a limit. Similarly, if a sequence is monotonically decreasing and bounded below, then it converges to a limit.