The Timpson Lyceum · Real Analysis (2020)

Lecture 4 (Real Analysis 2020)

Overview

This lecture closes out the foundational toolkit of real analysis and then opens the central idea of the course: convergence. It first nails down absolute value and proves the inequalities that govern it, culminating in the triangle inequality. It then collects several algebraic identities (difference of powers, the geometric sum, factorials, binomial coefficients, Pascal's identity, and the binomial theorem) that recur throughout later proofs. Finally it defines sequences and series, motivates infinite sums with Zeno's paradox, and states the precise $\varepsilon$ – $N$ definition of convergence — the bedrock for limits, continuity, differentiation, and integration. Master the convergence definition; everything in the following chapter is a special case of it.

Key Concepts

Density (recap). A set $S$ is dense in $\mathbb{R}$ if every open interval $(a,b)$ contains an element of $S$ . The rationals $\mathbb{Q}$ and the irrationals are both dense in $\mathbb{R}$ ; the integers are not (e.g. $(0.5, 0.6)$ contains no integer). The density proofs lean on the Archimedean property: for any $\delta > 0$ there is a natural number $n$ with $\tfrac{1}{n} < \delta$ .

Absolute value. Defined piecewise: $|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$ When $x < 0$ , the expression $-x$ is a positive number — this sign-flip is the usual source of confusion in proofs.

Bounding inequality. For $c, d \in \mathbb{R}$ : $|c| \le d \iff -d \le c \le d.$ This is proved both directions, splitting into the cases $c \ge 0$ and $c < 0$ .

Sign-sandwich. For every $x \in \mathbb{R}$ : $-|x| \le x \le |x|.$

Triangle inequality. For all $a, b \in \mathbb{R}$ : $|a + b| \le |a| + |b|.$

Interval form of a bound. For real $a$ and $r > 0$ , these are three ways of saying the same thing: $|x - a| < r \iff -r < x - a < r \iff x \in (a - r,\, a + r).$ This is exactly the shape used to define a limit.

Difference of powers. $a^n - b^n = (a - b)\big(a^{n-1} + a^{n-2}b + \cdots + a\,b^{n-2} + b^{n-1}\big)$ . Useful for factoring.

Geometric sum. For $r \ne 1$ : $\;1 + r + r^2 + \cdots + r^n = \dfrac{r^{n+1} - 1}{r - 1}$ .

Factorial. $0! = 1$ , and for $n \in \mathbb{N}$ , $\,n! = n \cdot (n-1)!$ .

Binomial coefficient. $\dbinom{n}{k} = \dfrac{n!}{(n-k)!\,k!}$ — the number of $k$ -element subsets of an $n$ -element set.

Pascal's identity. $\dbinom{n}{k} = \dbinom{n-1}{k-1} + \dbinom{n-1}{k}$ .

Binomial theorem. $(a+b)^n = \sum_{k=0}^{n} \dbinom{n}{k} a^{n-k} b^{k}$ .

Sequence. A real-valued function whose domain is $\mathbb{N}$ , written $(a_n)$ , where $a_n$ is the $n$ th term. Order matters. Sequences may be given by a formula (e.g. $a_n = \tfrac{1}{n}$ ) or recursively (like the factorial).

Series. Given a sequence $(c_n)$ , the associated series is the new sequence of partial sums $(s_n)$ where $s_n = \sum_{k=1}^{n} c_k$ . A series is itself a sequence. (Zeno's paradox dissolves once you see that an infinite sum of distances can be finite: $\tfrac12 + \tfrac14 + \tfrac18 + \cdots = 1$ .)

Convergence ( $\varepsilon$ – $N$ definition). A sequence $(a_n)$ converges to $a$ provided that for every $\varepsilon > 0$ there exists an index $N \in \mathbb{N}$ such that $|a_n - a| < \varepsilon \quad \text{for all } n \ge N.$ Intuitively: no matter how small a tolerance band $(a - \varepsilon,\, a + \varepsilon)$ you choose around $a$ , the sequence eventually enters that band and stays inside it forever. The whole art of a convergence proof is finding $N$ in terms of $\varepsilon$ — and the Archimedean property is what produces it.

Worked Example

Prove that $\dfrac{2}{n^2} + \dfrac{4}{n} + 3$ converges to $3$ .

Fix any $\varepsilon > 0$ . We must produce an index $N$ so that $\left|\left(\tfrac{2}{n^2} + \tfrac{4}{n} + 3\right) - 3\right| < \varepsilon$ for all $n \ge N$ . Do scratch work to bound the tail, then choose $N$ via the Archimedean property.

\begin{aligned} \left|\tfrac{2}{n^2} + \tfrac{4}{n} + 3 - 3\right| &= \left|\tfrac{2}{n^2} + \tfrac{4}{n}\right| \;\le\; \tfrac{2}{n} + \tfrac{4}{n} \;=\; \tfrac{6}{n} \\ \tfrac{6}{n} < \varepsilon &\iff \tfrac{1}{n} < \tfrac{\varepsilon}{6} \;\;\xRightarrow{\text{Archimedean}}\;\; \exists\, N \in \mathbb{N},\; \tfrac{1}{N} < \tfrac{\varepsilon}{6} \\ \text{then for all } n \ge N:\quad \left|\tfrac{2}{n^2} + \tfrac{4}{n} + 3 - 3\right| &\le \tfrac{6}{n} \le \tfrac{6}{N} < \varepsilon. \end{aligned}

Since $\varepsilon > 0$ was arbitrary, the sequence converges to $3$ . (The simpler companion proof, that $\tfrac{1}{n} \to 0$ , runs identically: choose $N$ with $\tfrac{1}{N} < \varepsilon$ , then $\left|\tfrac{1}{n} - 0\right| = \tfrac{1}{n} \le \tfrac{1}{N} < \varepsilon$ for all $n \ge N$ .)

Summary

$|x|$ equals $x$ when $x \ge 0$ and $-x$ when $x < 0$ ; in proofs, $-x$ denotes a positive quantity whenever $x$ is negative.
The core bound $|c| \le d \iff -d \le c \le d$ and the sandwich $-|x| \le x \le |x|$ combine to give the triangle inequality $|a+b| \le |a| + |b|$ .
$|x - a| < r$ , $\,-r < x - a < r$ , and $x \in (a-r,\,a+r)$ all say the same thing — the template for the limit definition.
Keep the algebraic identities handy: difference of powers, the geometric sum $\tfrac{r^{n+1}-1}{r-1}$ , factorials, $\binom{n}{k}$ , Pascal's identity, and the binomial theorem.
A sequence is a function $\mathbb{N} \to \mathbb{R}$ ; a series is the sequence of its partial sums $s_n = \sum_{k=1}^n c_k$ .
$(a_n) \to a$ means: for every $\varepsilon > 0$ there is an $N$ with $|a_n - a| < \varepsilon$ for all $n \ge N$ . Convergence proofs amount to building $N$ from $\varepsilon$ , usually through the Archimedean property.