Epsilon-delta definition of a limit

Statement

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, let $a$ be a limit point of $D$, and let $L$ be a real number. Then $\lim_{x \to a} f(x) = L$ if and only if the following holds. For each $\varepsilon > 0$ there is $\delta > 0$ such that $|f(x) - L| < \varepsilon$ for all $x \in D$ with $0 < |x - a| < \delta$.

Remarks

Each open interval around a point contains an interval $(L - \varepsilon, L + \varepsilon)$, and each such interval is open. This gives the equivalence with the definition. Many texts take this condition as the definition of the limit. Mathlib states a form for metric spaces as Metric.tendsto_nhds_nhds.

Explanations

L1 human

For each error tolerance $\varepsilon$, we can force $f(x)$ within $\varepsilon$ of $L$. We do this by taking $x$ within a suitable distance $\delta$ of $a$.