[4.2.3] Sequential Criterion for Functional Limits:
Let \(f: A \rightarrow \mathbb{R}\) and let \(c\) be a limit point of \(A\), the following two statements are equivalent:
Let \(f: A \rightarrow \mathbb{R}\) and let \(c\) be a limit point of \(A\), the following two statements are equivalent:
- \(\lim\limits_{x \to c} f(x) = L\).
- For all sequences \((x_n) \subseteq A\) satisfying \(x_n \neq c\) and \((x_n) \rightarrow c\), it follows that \(f(x_n) \rightarrow L\).
Definition of functional limits
Proof
TODO
References