Prove that every convergent sequence of real numbers is bounded. Moreover, if ( an ) is converge

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Prove that every convergent sequence of real numbers is...

Key Concepts

Infimum and Supremum

Infimum (greatest lower bound) and supremum (least upper bound) are key concepts in understanding the bounds of a set of real numbers. In the context of convergent sequences, showing that the limit lies between the infimum and supremum of the sequence reinforces that the limit is an accumulation point of the sequence and aligns with the order structure of the real numbers.

Convergent Sequence

A convergent sequence is one where the terms approach a specific real number, called the limit, as the index increases. Formally, for every ? > 0 there exists an N such that for all n ? N, the distance between a_n and the limit L is less than ?. This concept is fundamental in real analysis as it formalizes the intuitive idea of stability of the sequence as it progresses.

Bounded Sequence

A sequence is bounded if there exists some real number M such that the absolute value of every term in the sequence is less than or equal to M. The significance in context is that convergence implies boundedness, meaning that if a sequence converges to a limit, its terms cannot be arbitrarily large or small, which assures that the sequence stays within fixed global bounds.

Epsilon Definition of Convergence

The ?-definition provides the precise language needed to prove properties of converging sequences. It states that for every ? > 0, there exists an index N beyond which all elements of the sequence are within ? of the limit. This definition is a critical tool in showing that the tail of any convergent sequence resides within a finite interval, thereby demonstrating boundedness.

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