[Understanding Analysis by Stephen Abbott] - [Chapter 2 Sequences and Series]

Definition 2.2.1 A sequence is a function whose domain is $N$.

Definition 2.2.3 Convergence of a Sequence A sequence $(a_n)$ converges to a real number $a$ if, for every positive number $\epsilon$, there exists an $N \in N$ such that whenever $n \leqslant N$ it follows that $|a_n - a| < \epsilon$

To indicate that $(a_n)$ converges to $a$, we usually write either $\lim a_n = a$ or $(a_n) \to a$. The notation $\lim_{n \to \infty} a_n$ is also standard.

Definition 2.2.4 Given a real number $a \in R$ and a positive number $\epsilon > 0$ the set $$ V_\epsilon(a) = \{x \in R : |x - a| < \epsilon\} $$ is called the $\epsilon$-neighborhood of $a$.

Notice that $V_\epsilon(a)$ consists of all of those points whose distance from $a$ is less than $\epsilon$. Said another way, $V_\epsilon(a)$ is an interval, centered at $a$, with radius $\epsilon$.

Definition 2.2.3B Convergence of a Sequence: Topological Version. A sequence $(a_n)$ converges to $a$ if, given any $\epsilon$-neighborhood $V_\epsilon(a)$ of $(a)$, there exists a point in the sequence after which all of the terms are in $V_\epsilon(a)$. In other words, every $\epsilon$-neighborhood contains all but a finite number of the terms of $(a_n)$

Theorem 2.2.7 (Uniqueness of Limits) Uniqueness of Limits. The limit of a sequence, when it exists, must be unique.

Definition 2.2.9 A sequence that does not converge is said to diverge.

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[Understanding Analysis by Stephen Abbott] - [Chapter 2 Sequences and Series] - [2.2 The Limit of a Sequence]