Proving Convergence of a Monotonic Increasing Sequence to Its Least Upper Bound

In the realm of real analysis, understanding the behavior of sequences is fundamental. One crucial aspect is to demonstrate that a monotonic increasing sequence converges to its least upper bound, also known as the supremum. This real analysis concept is not only theoretical but also pivotal for various mathematical proofs and applications.

Definitions and Key Concepts

To begin, let's define the terms involved in this discussion:

Monotonic Increasing Sequence: A sequence (a_n) is called monotonic increasing if (a_n leq a_{n 1}) for all (n). Least Upper Bound (Supremum): The least upper bound of a set (S subseteq mathbb{R}) is the smallest number (L) such that (L geq s) for all (s in S).

Steps to Show Convergence to the Supremum

Existence of a Supremum: Since a monotonic increasing sequence is bounded above, it has a least upper bound denoted by (L sup{a_n : n in mathbb{N}}). Behavior of the Sequence: For any epsilon ((epsilon > 0)), find an integer (N) such that for all (n geq N), (|a_n - L| . This is equivalent to showing:

(L - epsilon leq a_n leq L epsilon)

This implies that the terms of the sequence are within any predefined distance (epsilon) of the supremum (L).

Upper Bound Property: By the definition of the supremum, for any epsilon ((epsilon > 0)), there exists some (a_k) in the sequence such that (L - epsilon leq a_k leq L). Therefore, (a_k) is within (epsilon) of (L) from below. Monotonicity: Since the sequence is increasing, for all (n geq k), (a_k leq a_n leq L). Thus, for all (n geq k), (|L - a_n| leq epsilon).

Concluding the Proof

Therefore, for any (epsilon > 0), there exists an integer (N) such that for all (n geq N), the terms of the sequence are sufficiently close to the supremum (L). Formally, this shows that (a_n) converges to (L), or equivalently, (lim_{n to infty} a_n L).

Summary

A monotonic increasing sequence that is bounded above converges to its least upper bound (L) because for any (epsilon > 0), we can find an index (N) such that for all (n geq N), the terms of the sequence are within (epsilon) of (L). This proof hinges on the upper bound property and the nature of monotonic sequences in real analysis.

This straightforward proof provides insights into the foundational concepts of real analysis and is of immense importance in mathematical discourse and applications.