Absolute Value, Bounded Sets, and Inequalities

The completeness axiom gives $\mathbb{R}$ its structure, but analysis is done with a second idea layered on top: distance. The absolute value converts the order on $\mathbb{R}$ into a measure of how far apart two numbers are, and its governing inequality, the triangle inequality, appears in essentially every estimate in the subject.

The absolute value as size

Think of $|x|$ as the size of $x$ with its sign discarded, and of $|x-y|$ as the distance between $x$ and $y$ on the number line. The basic algebra follows straight from the two cases.¹

The fourth item is the one used most: it converts a statement about a size into a pair of ordinary inequalities, and back. In particular $|x-a|<ϵ$ says exactly $a-ϵ<x<a+ϵ$, i.e. $x$ lies within $ϵ$ of $a$. The definition of a limit is written in exactly this form.

Neighborhoods and the max/min identities

The interval interpretation of $|x-a|<ϵ$ names a set used constantly in later lessons: the $ϵ$-neighborhood of $a$, the open interval $(a-ϵ,a+ϵ)$ of points within $ϵ$ of $a$. Convergence, continuity, and limits are all phrased as for every $ϵ$-neighborhood of the target, eventually everything lands inside it. The non-strict version, $|x-a|\le ϵ\leftrightarrow a-ϵ\le x\le a+ϵ$, describes the closed interval $[a-ϵ,a+ϵ]$. Reading a distance bound as membership in an interval, and back, is a move made without comment from here on.

The absolute value also gives closed-form expressions for the larger and smaller of two numbers. Adding or subtracting the gap $|x-y|$ to the average $\frac{x+y}{2}$ selects the maximum or the minimum:

If $x\ge y$ then $|x-y|=x-y$, and the first formula collapses to $x$, the second to $y$; the case $x<y$ is symmetric. These identities make $\max$ and $\min$ ordinary algebraic functions of $x$ and $y$, which is why the maximum of two continuous functions is again continuous — a fact used when splitting a function into its positive and negative parts for integration.

The triangle inequality

The triangle inequality compares the size of a sum with the sum of the sizes.

The name comes from the geometry it generalizes: the straight path from $0$ to $x+y$ is no longer than the detour through an intermediate point. Written with distances, $|a-c|\le |a-b|+|b-c|$ — the direct distance from $a$ to $c$ is at most the trip via $b$. Two corollaries are used almost as often as the theorem.¹

The reverse form bounds the size of a difference below by the difference of sizes; it is what guarantees that if $x$ and $y$ are close, then $|x|$ and $|y|$ are close, and so the absolute value is a continuous function. To prove it, apply the triangle inequality to $a=(a-b)+b$, obtaining $|a|-|b|\le |a-b|$, then swap $a$ and $b$ to get $|b|-|a|\le |a-b|$; together these give the reverse form. The difference form is the plain triangle inequality with $y$ replaced by $-y$.

A worked estimate

Bounding an expression over an interval is the routine use of these inequalities.

Analysis proofs are full of deliberately loose bounds like this, because a crude inequality that is easy to write often suffices to drive an error term to zero.

Bounded functions

The estimate above says the function $x\mapsto x^2-9x+1$ stays below a fixed number on $[-1,5]$. That is boundedness.

The same polynomial is bounded on $[-1,5]$ but unbounded on all of $\mathbb{R}$, so boundedness is a property of the function together with its domain. The sup/inf vocabulary transfers from sets to functions by applying it to the range $f(D)=\{f(x):x\in D\}$:

A bounded function is one whose range is a bounded set, so ${\sup }_{x}f(x)$ and ${\inf }_{x}f(x)$ are finite real numbers. Geometrically the graph is trapped in a horizontal band: it never rises above the line $y=\sup f$ nor drops below $y=\inf f$, and neither line can be moved toward the graph and still bound it.

Comparing two functions pointwise passes to their suprema, but only in one direction.

The variables on the two sides are independent (the $x$ in ${\sup }_{x}f(x)$ and the one in ${\sup }_{x}g(x)$ range separately), so the conclusion reads ${\sup }_{x}f(x)\le {\sup }_{y}g(y)$. A common error is to strengthen the conclusion to ${\sup }_{x}f(x)\le {\inf }_{y}g(y)$. That is false under the pointwise hypothesis $f(x)\le g(x)$; it holds only under the stronger, separated hypothesis $f(x)\le g(y)$ for all pairs $x,y\in D$, the same distinction between pointwise and uniform comparison seen in the separated-sets proposition.

The difference is genuine: take $f=g$ any nonconstant bounded function. Pointwise $f(x)\le g(x)$ holds with equality, and $\sup f=\sup g$, but $\sup f>\inf g$, so the separated conclusion fails. Keeping the two hypotheses apart prevents a class of mistakes that recurs whenever suprema are compared.

The triangle inequality chains these distance statements: if $x$ is near $y$ and $y$ is near $z$, then $x$ is near $z$, with the errors adding. Every convergence proof in the sequences chapter rests on this.