Sequential_Criterion_for_Functional_Limits
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Definition
Given a function $f: A \to \mathbb{R}$ and a limit point $c$ of $A$, the following are equivalent:
- $\lim_{x \to c} f(x) = L$
- For all sequences $(x_n) \subseteq A$ with $x_n \neq c$ and $(x_n) \to c$, it follows that $f(x_n) \to L$
Proof
(1 $\Rightarrow$ 2) Assume $\lim_{x \to c} f(x) = L$. Given $\epsilon > 0$, there exists $\delta > 0$ such that $0 < |x - c| < \delta$ implies $|f(x) - L| < \epsilon$.
If $(x_n) \to c$ with $x_n \neq c$, there exists $N$ such that $n \geq N$ implies $|x_n - c| < \delta$. Thus $|f(x_n) - L| < \epsilon$ for $n \geq N$.
(2 $\Rightarrow$ 1) Contrapositive: If (1) fails, there exists $\epsilon_0 > 0$ such that for all $\delta > 0$, some $x$ satisfies $0 < |x - c| < \delta$ but $|f(x) - L| \geq \epsilon_0$.
Taking $\delta = 1/n$, we construct $(x_n) \to c$ with $|f(x_n) - L| \geq \epsilon_0$, so (2) fails.