Foundations — Real analysis — rigorous epsilon-delta, metric spaces

Before you can read a single line of the parent note, you need the alphabet it is written in. This page builds every symbol from nothing. Read top to bottom: each item uses only things defined above it.

1. Sets and membership — the containers

Some named pens we reuse constantly:

• $\mathbb{R}$ — the real numbers: every point on an infinite ruler, including fractions, decimals that never end, $\sqrt{2}$, $\pi$. Picture a horizontal line with no gaps.
• ${\mathbb{R}}^n$ — lists of $n$ real numbers, like $(x_1,x_2)$. For $n=2$ this is the flat plane (a sheet of paper); for $n=3$, space.

Why the topic needs this. A limit is a statement about a function's inputs $D$ and its outputs. Before we can talk about "getting close," we must first say which arena the points live in. That arena is a set.

2. Absolute value $|\cdot |$ — the ruler that measures gaps

The reason absolute value dominates analysis is the next idea:

Look at the figure. The two points $x$ and $a$ sit on the ruler; the orange bracket underneath is $|x-a|$, the length of the gap. Notice it is the same length whether you walk left-to-right or right-to-left — that symmetry is exactly why $|x-a|=|a-x|$.

Why the topic needs this. Every $\epsilon$–$\delta$ statement is built from $|x-a|$ (input gap) and $|f(x)-L|$ (output gap). Absolute value is the measuring tape of the real line — and in section 4 of the parent it gets generalised into the abstract distance $d$.

3. Inequalities and intervals — "within a tolerance"

The blue band in the figure is the set of all $x$ with $|x-a|<\delta$. The little hollow dot at $a$ shows what $0<|x-a|$ does: it punctures the centre, removing the single point $x=a$. So $0<|x-a|<\delta$ is "the blue band with its centre pin-hole removed."

Why the topic needs this. "Within $\epsilon$" and "within $\delta$" — the two rulers of the whole game — are inequalities on absolute values. No inequalities, no way to phrase "close."

4. Functions and the symbols $f$, $L$, $a$ — the machine

The three lead characters of the limit story:

• $a$ — the input value we are approaching (a spot on the horizontal ruler).
• $L$ — the output value we claim the machine's results are heading toward.
• $f(x)$ — the actual output when the input is $x$.

In the figure the curve is $y=f(x)$. As inputs $x$ (bottom axis) slide toward $a$, watch the outputs $f(x)$ (left axis) climb toward the height $L$. The horizontal orange band of half-width $\epsilon$ around $L$ is the target tube; the vertical blue band of half-width $\delta$ around $a$ is the safe zone. The limit statement is: whenever an input lands in the blue zone, its output lands in the orange tube.

Why the topic needs this. The limit ${\lim }_{x\to a}f(x)=L$ is a sentence about exactly these three symbols. Without a clear picture of input-machine-output, the definition is just noise.

5. The quantifiers $\forall$ and $\exists$ — the grammar of the game

Why the topic needs this. The entire definition of a limit is one quantified sentence. Reading it wrong (or in the wrong order) means reading a different, false statement.

6. The Greek letters $\epsilon$ and $\delta$ — the two tolerances

Why the topic needs this. These two letters are the definition. Everything in sections 1–5 exists so that "$0<|x-a|<\delta ⟹|f(x)-L|<\epsilon$" can be read fluently.

7. From line to plane — distance $d$ and the metric leap

The symbol $\sum$ (capital sigma) that appears in the metric table just means "add these up": ${\sum }_{i=1}^n{a}_i=a_1+a_2+\cdots +a_n$ — the running total, needed the moment we leave the one-dimensional ruler and add up gaps in several directions (see Sequences and Series).

Prerequisite map

Equipment checklist