Foundations — Epsilon-delta definition of a limit — formal proofs

This page assumes you have seen nothing. Before you can even read the line $\forall \epsilon >0\exists \delta >0\text{s.t.}0<|x-a|<\delta \Rightarrow |f(x)-L|<\epsilon ,$ you must know what each mark means and what each looks like. We build them one at a time, in an order where nothing is used before it is earned. When we finish, that line will read like a plain English sentence.

1. A number line, and the point $a$

Start with the simplest picture in all of mathematics: a straight horizontal line, with numbers marked on it. Left is smaller, right is bigger.

We will keep circling back to one special place on this line: the value the input is heading toward. We call it $a$.

Why the topic needs it: the whole game is about behaviour near $a$. Without a fixed landmark to approach, "close" has no meaning.

2. The variable $x$ and the function $f$

So we actually have two lines: a horizontal input line (where $x$ and $a$ live) and a vertical output line (where $f(x)$ lands). The graph is the picture that connects them.

Why the topic needs it: limits are statements about what the output $f(x)$ does while the input $x$ moves toward $a$.

3. The proposed limit $L$

Note carefully: $L$ is a guess we must verify. The definition never assumes $L$ is right; it gives a way to check.

4. Distance, and why we need the absolute value $|\cdot |$

Here is the first real tool. We keep saying "close." Close means "small distance." So we need a clean way to measure distance between two numbers.

If $x$ is at $5$ and $a$ is at $3$, the gap is $2$. If $x$ is at $1$ and $a$ is at $3$, the gap is also $2$ — but $1-3=-2$, a negative number. Distance can never be negative, so we throw away the sign.

Why the topic needs it: both "$x$ close to $a$" and "$f(x)$ close to $L$" become the two honest inequalities $|x-a|<\text{small}$ and $|f(x)-L|<\text{small}$.

5. Inequalities $<$, $\le$, and the "band" they draw

The key move: an inequality about distance draws a band (an interval) on the line.

Why are they the same? $|x-a|<\delta$ says "the gap between $x$ and $a$ is under $\delta$," which means $x$ can be at most $\delta$ to the right ($x<a+\delta$) or at most $\delta$ to the left ($x>a-\delta$). Two directions, one band.

Why the topic needs it: the entire epsilon–delta definition is two bands (one on the output, one on the input) and a promise linking them.

6. The two special small numbers: $\epsilon$ and $\delta$

Now name the two band half-widths. Both are Greek letters, both always strictly positive (a band of width zero would contain no room to move).

Why the topic needs it: these two are the whole vocabulary of "how close." $\epsilon$ is the challenge; $\delta$ is the reply.

7. The strict "$0<$": excluding the point itself

Look again at $0<|x-a|<\delta$. The extra left piece $0<|x-a|$ says the distance is strictly more than zero — so $x\ne a$.

Why the topic needs it: it is what makes limits able to talk about holes and gaps a plain "plug in $a$" cannot.

8. The arrow $\to$ and the symbol $\lim$

Why the topic needs it: it is the compact name for the very thing the $\epsilon$–$\delta$ machinery is defining.

9. The quantifiers $\forall$ and $\exists$ — and their order

These two symbols are what make the definition a game, and getting their order right is the single most important idea on this page.

The definition's spine is: $\underset{\text{whatever demand}}{\underbrace{\forall \epsilon >0}}\underset{\text{I can answer}}{\underbrace{\exists \delta >0}}\text{s.t. (the promise holds).}$

Why the topic needs it: the entire "challenge–response duel" and the proof recipe ("Let $\epsilon >0$. Choose $\delta =\dots$") is nothing but reading these quantifiers in order.

10. Implication $\Rightarrow$ — the promise itself

In the definition, $0<|x-a|<\delta \Rightarrow |f(x)-L|<\epsilon$ is the promise: if the input lands inside your $\delta$-band (and isn't $a$), then the output is guaranteed inside the $\epsilon$-band. A promise is broken only if the input obeyed but the output escaped.

Why the topic needs it: it is the exact link that turns "input close" into "output close" — the heart of the whole definition.

11. Putting the sentence together

Now every mark is earned. Re-read the definition as plain English:

$\forall \epsilon >0\exists \delta >0:0<|x-a|<\delta \Rightarrow |f(x)-L|<\epsilon$

Not one symbol is now mysterious. You are ready for the proofs in Epsilon-delta definition of a limit — formal proofs.

Prerequisite map

Each foundation feeds the next; the two bands and the quantifier order meet at the implication, which is the limit definition.

Equipment checklist

Test yourself — each line hides its answer.