Introductory Real Analysis: Completeness, Sequences, and Continuity

Core idea

Real analysis begins with the real numbers and asks why approximation processes have limits. Algebra supplies addition, multiplication, and order. Analysis adds a decisive structural fact: the real line has no gaps.

These notes develop that idea through three connected themes:

order, upper bounds, and completeness;
sequences and the Cauchy criterion;
continuity as preservation of limits.

Most substantive mathematical terms in the prose are expandable knowls. Open one when you want a definition, theorem statement, example, or prerequisite without leaving the lecture.

1. Order and completeness

The real numbers form an ordered field. Their order is governed by the order axioms, while absolute value turns order into distance:

d(x,y)=|x-y|.

An interval contains every real number lying between any two of its points. Intervals express the geometric idea that the real line is continuous rather than discrete.

The rational numbers are also ordered and dense, but they are not complete. Consider

S=\{q\in\mathbb Q:q>0\text{ and }q^2<2\}.

This set is nonempty and bounded above in $\mathbb Q$ , but it has no supremum in $\mathbb Q$ . The missing boundary would have to be $\sqrt 2$ .

Definition: supremum

Let $A\subseteq\mathbb R$ be nonempty and bounded above. A number $s\in\mathbb R$ is the supremum of $A$ when:

$s$ is an upper bound of $A$ ; and
every upper bound $u$ of $A$ satisfies $s\le u$ .

Thus the supremum is the least upper bound. It need not belong to the set, so it need not be a maximum.

The least-upper-bound theorem

Every nonempty subset of $\mathbb R$ that is bounded above has a supremum in $\mathbb R$ . This least-upper-bound theorem is one standard form of the completeness axiom.

Example: constructing $\sqrt 2$

Let

A=\{x\in\mathbb R:x\ge0\text{ and }x^2<2\}.

The set is nonempty because $1\in A$ , and it is bounded above because $2$ is an upper bound. Completeness gives $s=\sup A$ . One proves that $s^2=2$ : if $s^2<2$ , a sufficiently small positive increment keeps the square below $2$ , contradicting that $s$ is an upper bound; if $s^2>2$ , a sufficiently small decrement remains an upper bound, contradicting the minimality of $s$ .

Completeness has produced a real number from an approximation process. That pattern recurs throughout analysis.

The Archimedean property

The Archimedean property says that the natural numbers are unbounded in the real numbers. Equivalently, for every $\varepsilon>0$ , some $n\in\mathbb N$ satisfies $1/n<\varepsilon$ . This lets discrete indices control arbitrarily fine estimates.

2. Sequences and limits

A sequence is a function $a:\mathbb N\to\mathbb R$ , usually written $(a_n)$ . We say that the sequence converges to $L$ if, for every $\varepsilon>0$ , there exists $N\in\mathbb N$ such that

n\ge N\quad\Longrightarrow\quad |a_n-L|<\varepsilon.

The number $L$ is the limit of the sequence. The definition separates the requested accuracy $\varepsilon$ from the stage $N$ after which that accuracy is guaranteed.

Example: a rational sequence

Consider

a_n=\frac{n}{n+1}=1-\frac{1}{n+1}.

Given $\varepsilon>0$ , choose $N>1/\varepsilon$ . If $n\ge N$ , then

|a_n-1|=\frac{1}{n+1}<\varepsilon.

Hence $a_n\to1$ . Once basic limits are known, the limit algebra for sequences combines them under addition, multiplication, and division when the limiting denominator is nonzero.

Every convergent sequence is a bounded sequence. The converse is false: $(-1)^n$ is bounded but does not converge. Nevertheless, boundedness guarantees partial convergence. The Bolzano–Weierstrass theorem says that every bounded sequence in $\mathbb R^k$ has a convergent subsequence.

A subsequence selects terms $a_{n_1},a_{n_2},\ldots$ with strictly increasing indices. It can reveal limiting behavior hidden by the full sequence. For example, the even and odd subsequences of $(-1)^n$ converge to different values, proving that the original sequence cannot converge.

3. Cauchy sequences

The definition of convergence mentions the proposed limit $L$ . Sometimes we can prove that approximations stabilize before knowing what their limit should be.

A sequence $(a_n)$ is a Cauchy sequence if, for every $\varepsilon>0$ , there exists $N\in\mathbb N$ such that

m,n\ge N\quad\Longrightarrow\quad |a_m-a_n|<\varepsilon.

The terms eventually become close to one another. The Cauchy criterion states that a real sequence converges if and only if it is Cauchy.

Proof idea

If $a_n\to L$ , the triangle inequality gives, for sufficiently large $m,n$ ,

|a_m-a_n|\le |a_m-L|+|a_n-L|<\varepsilon.

Conversely, a Cauchy sequence is bounded. The Bolzano–Weierstrass theorem supplies a convergent subsequence, and the Cauchy property forces the entire sequence toward that subsequential limit. This reverse implication is where completeness enters.

In a general metric space, not every Cauchy sequence must converge. A metric space in which every Cauchy sequence converges is called a complete metric space. Thus completeness converts internal consistency of approximations into existence of a limit.

4. Continuity preserves limits

Let $f:\mathbb R\to\mathbb R$ . The function is continuous at a point $x$ if, for every $\varepsilon>0$ , there exists $\delta>0$ such that

|y-x|<\delta\quad\Longrightarrow\quad |f(y)-f(x)|<\varepsilon.

The sequential criterion for continuity says that $f$ is continuous at $x$ exactly when

x_n\to x\quad\Longrightarrow\quad f(x_n)\to f(x).

Continuity therefore transports convergent input approximations to convergent output approximations.

Example: continuity of $f(x)=x^2$

Fix $x\in\mathbb R$ . For $y$ near $x$ ,

|y^2-x^2|=|y-x|\,|y+x|.

First require $|y-x|<1$ , which implies $|y+x|\le2|x|+1$ . It is then enough to choose

\delta=\min\left\{1,\frac{\varepsilon}{2|x|+1}\right\}.

This is a common estimate pattern: impose a coarse local bound first, then tune the remaining factor to $\varepsilon$ .

5. A global consequence of continuity

The intermediate value theorem says that if a continuous function $f:[a,b]\to\mathbb R$ takes values on opposite sides of a number $u$ , then it takes the value $u$ somewhere in the interval.

Apply the theorem to the polynomial $f(x)=x^2-2$ . Since $f(1)<0$ and $f(2)>0$ , there is $c\in(1,2)$ with $c^2=2$ . Earlier, completeness produced $\sqrt2$ as a supremum; now continuity produces it as a zero. These are two manifestations of the gap-free structure of the real numbers.

6. Exercises

Prove directly from the definition of convergence that $(2n+1)/(n+3)\to2$ .
Prove that every convergent sequence is bounded.
Prove that a sequence has at most one limit.
Show directly that every convergent sequence is a Cauchy sequence.
Use the sequential criterion for continuity to prove that $f(x)=3x-5$ is continuous.
Let $f(x)=x^3+x-1$ . Use the intermediate value theorem to prove that $f$ has a zero in $(0,1)$ .
Identify exactly where completeness is used in the proof of the Cauchy criterion.

Takeaway

The least-upper-bound property, convergence of Cauchy sequences, and preservation of limits by continuous functions are not isolated techniques. They are different expressions of one organizing principle: the real numbers contain the limits demanded by consistent approximation.