Convergence of sequences: definitions and foundations

The rigorous definition of the limit

Introduction: Beyond Intuition

A sequence (u_n) converges to a limit l as n becomes large. However, ‘as n becomes large’ and ‘approaches’ remain vague. The rigorous definition, derived from Cauchy and Weierstrass, replaces intuition with a verifiable quantitative criterion.

The epsilon-N definition

We say that (u_n) converges to l if:

for every epsilon > 0, there exists an integer N such that for every n >= N, |u_n - l| <= epsilon

In other words: however small the margin of error ε that we allow ourselves, all terms of the sequence eventually fall within the interval [l – ε, l + ε], from a certain order N which depends on ε.

Detailed example

Let $u_n = 1/n$ . Let us show that $u_n \to 0$ . Let $\epsilon > 0$ be a fixed value. We are looking for $N$ such that, for $n \geq N$ , |1/n - 0| ≤ ε, i.e. 1/n ≤ ε, i.e. n ≥ 1/ε. It suffices to take N equal to the integer part of (1/ε) plus 1. The definition is then satisfied for all n ≥ N.

Common pitfalls

A common pitfall is to believe that $N$ must be the same for every $\epsilon$ . On the contrary, $N$ depends on $\epsilon$ : the smaller $\epsilon$ is, the larger $N$ generally is. Another pitfall is to reverse the order of the quantifiers. ‘For all epsilon’ always comes first, ‘there exists an N’ second; this order is essential and must never be reversed.

Notation

We write lim(n→∞) u_n = l, or more simply u_n → l.