Week 1, Lecture 2: Numbers, Metric Properties, and Completeness

Finding Your Footing: Numbers, Fractions, and Proofs

Welcome to Lecture 2! In our previous session, we established the grammatical foundation of sets and mappings. Today, we turn our attention to numbers—objects you have been working with since primary school. Yet, as we examine them through the lens of real analysis, familiar concepts will take on a deeper, more rigorous character.

You might wonder why we need formal proofs to establish things that feel intuitively obvious—such as the fact that between any two fractions, there is always another fraction, or that $\sqrt{2}$ cannot be written as a simple ratio of integers. After all, if our calculators and geometric intuition already tell us these things are true, why labor through the logic?

The reason is that intuition alone can hide subtle traps. The ancient Greeks were profoundly shocked to discover that the diagonal of a unit square ($\sqrt{2}$) slips entirely through the cracks of the rational numbers, revealing invisible "holes" in the number line. Proving these properties from first principles gives us absolute certainty, ensuring that when we eventually build limits and calculus upon these numbers, our foundation never shakes.

If constructing proofs by contradiction or working through algebraic inequalities feels challenging at first, take a deep breath—that is completely normal. Mathematical rigor is a craft acquired one careful step at a time. Take your time with the examples, explore the arguments, and remember that every expert analyst started exactly where you are now.

The Four Pillars of Peano's Axioms

1. The Root: $0$ is a natural number.
2. Successors (No Branching): Each natural number has exactly one unique successor.
3. Uniqueness (No Loops): $0$ is not the successor of any natural number, and every other natural number is the successor of exactly one predecessor.
4. Principle of Mathematical Induction: A statement is true for all natural numbers if we can prove it holds for $n = 0$ (the base case) and that assuming it holds for $n-1$, it necessarily holds for $n$.

When we patch the gaps in the rational numbers $\mathbb{Q}$ to build the real numbers $\mathbb{R}$, we want to make sure our new number line behaves nicely. We don't want arithmetic to act unpredictably or inequalities to contradict themselves.

To guarantee this, mathematicians define $\mathbb{R}$ as a complete ordered field. That sounds like intimidating vocabulary, but it simply means $\mathbb{R}$ is built on three common-sense promises:

1. It's a "Field" (The Arithmetic Promise): You can add, subtract, multiply, and divide freely, and all the familiar rules of algebra work. The arithmetic playground always plays fair. (Explore a deep dive on Groups, Rings, and Fields →)
2. It's "Ordered" (The Direction Promise): Numbers have a strict, consistent left-to-right direction. Every number is either smaller than, equal to, or greater than any other number, and inequalities behave predictably when you shift or scale them.
3. It's "Complete" (The No-Holes Promise): Remember the invisible pinprick hole we found at $\sqrt{2}$ in the rational numbers ($\mathbb{Q}$)? Completeness means there are zero holes left on the line. Every set of numbers that has an upper ceiling has a tightest possible ceiling sitting right inside the system, plugging every gap permanently.

Let's break down how these three promises translate into everyday rules you can trust:

In everyday speech, when someone says "take the absolute value," the rule of thumb you probably learned was simply: "throw away the minus sign." For example, $|5| = 5$ and $|-5| = 5$.

While that rule of thumb is fine for quick mental arithmetic, real analysis asks you to look at absolute value through a much more powerful lens: geometry and physical distance. If you place a tape measure between two marks on a workbench, the distance is never negative, regardless of whether you measure from left to right or from right to left.

1. How Absolute Value is Defined (The Split-Formula)

To express this rigorously without relying on informal words like "drop the sign", mathematics defines the absolute value function $|x|: \mathbb{R} \to \mathbb{R}$ piecewise:

Notice what happens when $x$ is negative: if $x = -4$, then $-x = -(-4) = +4$. The extra minus sign cancels out the negativity, guaranteeing the final output is always non-negative.

2. Distance Between Two Points along the Line

When you measure the distance between any two coordinates $a$ and $b$ on the number line, you simply take the difference between them inside an absolute value:

Because $|a - b| = |b - a|$, distance is completely symmetric: the distance from Alice at $2$ to Bob at $6$ is $|2 - 6| = |-4| = 4$, and the distance from Bob to Alice is $|6 - 2| = |4| = 4$.

3. Measuring "Closeness": $\varepsilon$-Neighbourhoods

Here is where real analysis truly begins. In calculus, when we say "$x$ is getting closer and closer to $a$," how do we write that with mathematical precision? We use the Greek letter epsilon ($\varepsilon$) to represent an arbitrarily small positive margin of error.

We say that $x$ is $\varepsilon$-close to $a$ if the distance between them is strictly less than $\varepsilon$:

Notation note: The double arrow $\iff$ stands for "if and only if" (logical equivalence). It simply means the statement on the left and the statement on the right are two different ways of saying the exact same truth: if one holds, the other automatically holds!

Geometrically, this creates an open safety buffer centered right at $a$, reaching $\varepsilon$ units to the left and $\varepsilon$ units to the right. In analysis, this open interval $(a - \varepsilon, a + \varepsilon)$ is called the $\varepsilon$-neighbourhood of $a$.

📐 Why is it called the "Triangle" Inequality?

Have you ever wondered why an algebraic rule about numbers on a flat line is named after a triangle?

In geometry, the shortest distance between two points is a direct straight line. If you start at point $A$ and want to go to $C$, walking directly along side $c$ will always be shorter than taking a detour through point $B$ (walking $a + b$). In any physical triangle, no single side can ever be longer than the other two sides combined:

On the real number line, this triangle is simply flattened into one dimension. If $a$ and $b$ pull in opposite directions (like $-2$ and $+4$), their cancellation shrinks the net result to $|-2 + 4| = 2$, which is strictly smaller than walking the full detour distance $|-2| + |4| = 6$. The only time equality holds ($|a + b| = |a| + |b|$) is when both numbers push in the exact same direction!

🔄 The Reverse Triangle Inequality: Setting the Floor

The forward triangle inequality gives you a ceiling: the direct journey $|a - b|$ can never be longer than $|a| + |b|$. But does it have a floor? Can the distance between two points shrink to zero if the two sticks have different lengths?

Of course not! If one stick has length $10$ and another stick has length $3$, the third side of any triangle connecting them must be at least $10 - 3 = 7$ units long, even if you fold them flat against each other. This is the Reverse Triangle Inequality:

The 2-Line Sneaky Proof: You don't need a whole new framework to prove this—it falls out of the forward triangle inequality with a clever substitution! Write $a$ as $(a - b) + b$:

Swapping the roles of $a$ and $b$ gives $|b| - |a| \le |a - b|$ as well. Putting those two bounds together wraps the difference into absolute values: $||a| - |b|| \le |a - b|$!

Up to this point, we have established arithmetic rules and measured distances between numbers. But we now arrive at the true crown jewel of introductory real analysis: Completeness.

In Section 3, we discovered that the rational numbers $\mathbb{Q}$ have missing coordinates like $\sqrt{2}$. We promised that the real numbers $\mathbb{R}$ "patch" all of these invisible pinpricks. But how do mathematicians describe an unbroken continuum without using hand-wavy words like "no holes"?

The answer lies in two deceptively simple concepts: ceilings (bounds) and tightest ceilings (suprema).

1. Ceilings and Floors: Upper and Lower Bounds

Imagine a collection of numbers sitting inside a set $S \subseteq \mathbb{R}$. Think of an upper bound as any ceiling that hangs above every single member of the set:

• Upper Bound: A number $K$ is an upper bound of $S$ if $x \le K$ for every $x \in S$. If such a number exists, we say $S$ is bounded above.
• Lower Bound: A number $k$ is a lower bound of $S$ if $k \le x$ for every $x \in S$. If such a number exists, we say $S$ is bounded below.
• Bounded Set: A set that has both a floor and a ceiling (bounded below and above simultaneously).

The Ceiling Ambiguity: Notice that ceilings are never unique! If you are standing in a room with a 3-metre ceiling, a 10-metre ceiling is also safely above your head, as is a 100-metre ceiling. If $K$ is an upper bound, any larger number $K' > K$ is automatically an upper bound too.

2. The Tightest Bounds: Supremum ($\sup$) and Infimum ($\inf$)

Since upper bounds can be unnecessarily tall (like choosing a ceiling a kilometer high), analysts ask: "What is the lowest, tightest possible ceiling you can place over the set without cutting into it?"

Similarly, the infimum ($t = \inf S$) is the greatest lower bound—the highest possible floor supporting the set from below.

The Crucial Difference: While $(0, 1)$ has no maximum, its supremum exists and is exactly $1$ ($\sup(0, 1) = 1$). A supremum acts as a generalized maximum: if the maximum exists, it equals the supremum; if the maximum does not exist, the supremum still pins down the exact border where the set ends!

3. The Axiom of Completeness: What Makes $\mathbb{R}$ Solid

Now we can formulate the bedrock principle that separates the rational numbers from the real numbers:

In plain language: if a set of real numbers has any ceiling at all, it is guaranteed to have an exact, tightest ceiling that belongs to the real number continuum.

4. The Archimedean Property: No Infinitely Large Numbers

Completeness guarantees there are enough numbers to plug every gap. But how do we ensure there aren't too many numbers (like mysterious infinitely large elements sitting beyond all natural counting)?

This is settled by the Archimedean Property of the real numbers:

In simple terms: no matter how tiny a step $x$ you take (say, a millimeter), and no matter how huge a mountain $y$ stands before you (the distance to the moon), taking sufficiently many repeated steps ($nx$) will eventually surpass $y$. The natural numbers $\mathbb{N}$ march without boundary across all of $\mathbb{R}$.

Ready to test your understanding?

Work through the progressive interactive problems to lock in these foundational concepts before moving on to Lecture 3.