---

State and Prove the Monotone Convergence Theorem

---

Introduction

The Monotone Convergence Theorem is an important result in real analysis. It guarantees that certain sequences — specifically, those that are monotonic and bounded — will always converge. This theorem is widely used in calculus and mathematical analysis for studying limits and series.

---

Statement of the Monotone Convergence Theorem (MCT)

Theorem:
Every sequence that is monotonic and bounded converges.

Specifically:

1. If a sequence {a_n} is monotonically increasing (that is, (a_1 \leq a_2 \leq a_3 \leq \dots)) and bounded above, it converges to its supremum (least upper bound).
2. If a sequence {a_n} is monotonically decreasing (that is, (a_1 \geq a_2 \geq a_3 \geq \dots)) and bounded below, it converges to its infimum (greatest lower bound).

---

Proof of the Monotone Convergence Theorem

Case 1: Monotonically Increasing Sequence

Let {a_n} be a sequence such that:

a_1 ≤ a_2 ≤ a_3 ≤ ... ≤ a_n ≤ ...

Also, assume the sequence is bounded above. That is, there exists a real number M such that:

a_n ≤ M for all n ∈ N,

where N is the set of natural numbers.

Since {a_n} is increasing and bounded above, the supremum (least upper bound) of the set {a_n : n ∈ N} exists. Let:

L = sup {a_n : n ∈ N}.

We need to show that:

lim (n → ∞) a_n = L.

Proof by contradiction:

Assume that the sequence {a_n} does not converge to L. Then, there must exist some ε > 0 such that:

|a_n - L| ≥ ε for infinitely many n.

However, since L is the supremum, all terms a_n must get arbitrarily close to L as n increases. This contradiction shows that:

lim (n → ∞) a_n = L.

Case 2: Monotonically Decreasing Sequence

Let {a_n} be a sequence such that:

a_1 ≥ a_2 ≥ a_3 ≥ ... ≥ a_n ≥ ...

Assume the sequence is bounded below. That is, there exists a real number m such that:

a_n ≥ m for all n ∈ N.

Since {a_n} is decreasing and bounded below, the infimum (greatest lower bound) of the set {a_n : n ∈ N} exists. Let:

L = inf {a_n : n ∈ N}.

We need to show that:

lim (n → ∞) a_n = L.

As in Case 1, we argue by contradiction. If the sequence does not converge to L, there must exist some ε > 0 such that:

|a_n - L| ≥ ε for infinitely many n.

This contradicts the fact that L is the infimum. Therefore:

lim (n → ∞) a_n = L.

---

Conclusion

The Monotone Convergence Theorem guarantees that any monotonic sequence (either increasing or decreasing) that is also bounded will converge to a real number. This result is essential in the study of limits and has wide applications in both theoretical and practical aspects of mathematics.