Convergence of Bounded Increasing Sequences

Prove that if an increasing sequence is bounded, then it converges to the supremum of the set of values that it takes on.

In this problem, we delve into one of the fundamental ideas in real analysis: the convergence of bounded increasing sequences. This concept is a crucial part of understanding how sequences behave, particularly within the scope of completeness in the real numbers. The problem asks us to prove that such a sequence converges to its supremum, which is an important result in the context of the least upper bound property of the real numbers.

To tackle this problem, we leverage the definition of a bounded sequence and the properties of increasing sequences. A bounded sequence is one where all terms lie within some fixed bounds, meaning it does not diverge to infinity. An increasing sequence is one where each term is greater than or equal to the one before it. Combining these two properties in the context of real analysis, we use the Completeness Axiom (also known as the Least Upper Bound Axiom), which stipulates that every non-empty set of real numbers that is bounded above has a least upper bound or supremum.

The strategy involves demonstrating that the supremum acts as the least upper bound in a precise sense, ensuring the sequence cannot surpass this limit. This is pivotal because it exemplifies how completeness in real numbers ensures that certain sequences always converge, a concept with far-reaching implications in analysis. This problem elegantly combines sequence behavior, bounds, and the foundational property of real numbers, offering a gateway into deeper analysis topics such as the Bolzano-Weierstrass Theorem and other convergence concepts.