Prove that if {an} is a convergent sequence, then to every...

Prove that if $\left\{a_{n}\right\}$ is a convergent sequence, then to every positive number $\varepsilon$ there corresponds an integer $N$ such that
$$
\left|a_{m}-a_{n}\right|<\varepsilon \quad \text { whenever } \quad m>N \quad \text { and } \quad n>N
$$

Key Concepts

Triangle Inequality

The triangle inequality is a fundamental tool in analysis that states for any three points, the absolute difference between the first and third is at most the sum of the differences between the first and second and the second and third. In the context of convergence and Cauchy sequences, it is used to relate the distance between any two terms of the sequence to their distances from the limit, thereby facilitating the proof that convergent sequences are Cauchy.

Epsilon-Definition (?-Definition) Argument

The ?-definition is a precise way to express the idea of convergence. It states that given any arbitrarily small positive number ?, there exists a point in the sequence beyond which all terms remain within an ? distance from the limit (or from each other in the case of Cauchy sequences). This definition is the backbone of proofs in analysis and is used to rigorously demonstrate the closeness properties of sequences.

Cauchy Sequence

A Cauchy sequence is one where the terms of the sequence become arbitrarily close to each other as the sequence progresses. More precisely, a sequence is Cauchy if for every positive number ?, there exists an index N such that for all m, n > N, the distance |a_m - a_n| between any two terms is less than ?. The significance of this concept lies in the fact that in complete spaces, every Cauchy sequence converges, and every convergent sequence is necessarily Cauchy.

Convergence of a Sequence

A sequence is said to converge if its terms get arbitrarily close to a specific number called the limit. Formally, a sequence {a_n} converges to a limit L if, for every positive number ?, there exists an integer N such that for all n > N, the absolute difference |a_n - L| is less than ?. This fundamental concept ensures that eventually, all terms of the sequence fall within any ?-neighborhood around L.

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