Limits of sequences

Definition 13

Limit of a sequence

$a(n) \to L$ means: for every $ε > 0$ there is $N$ such that $|a(n) - L| < ε$ for all $n \ge N$ .

Read the quantifiers in order: $\forall ε \exists N \forall n \ge N$ . Somebody hands you an $ε$ , however small. You must answer with an $N$ , which may depend on $ε$ .

To prove a limit, begin with “Let $ε > 0$ be arbitrary.” Then choose $N$ and check the inequality.