Unit 1

What calculus really means

The real numbers, limits, the derivative and the integral, made exact.

6 chapters written · 32 definitions · 48 theorems · builds on Unit 0

The story

School calculus works with limits it never defines. This unit defines them. The starting point is a property of the real numbers that school never states: every bounded set has a least upper bound (1.1). Every theorem in the unit is that one fact, used carefully.

From it come limits of sequences with $\varepsilon$ and $N$ (1.2), limits of functions and continuity with $\varepsilon$ and $\delta$ (1.3), the derivative as the best straight-line approximation (1.4), the integral as a squeeze between upper and lower sums (1.5), and series, where infinitely many terms add up to a number (1.6).

The same pattern repeats in every chapter: say what "close" means with a number, then prove that closeness can be forced. Unit 4 repeats it with distance in any space, and MATH 521 starts where this unit ends.

Chapters

The theorem chain

Every theorem in the unit, in the order it is proved. Each one is used by the ones after it. The label says whether it is on the 375/376 exams (T), an Apostol extra (A), or beyond both (B).

1.1 The real numbers and completeness

Theorem · B Approximation property of the supremum
Theorem · B The reals are Archimedean The natural numbers are not bounded above in $\mathbb R$ .
Theorem · B The rationals are dense Between any two real numbers there is a rational number.

1.2 Sequences and limits, from zero

Theorem · B Limits are unique A sequence cannot converge to two different limits.
Theorem · B Convergent sequences are bounded If $a_n \to L$ , some single number bounds every $|a_n|$ .
Theorem · B Algebraic limit laws If $a_n \to a$ and $b_n \to b$ , then $a_n + b_n \to a + b$ , $c\,a_n \to ca$ , $a_n b_n \to ab$ , and $a_n/b_n \to a/b$ when $b \ne 0$ .
Theorem · B Monotone convergence theorem An increasing sequence that is bounded above converges, to the supremum of its terms.
Theorem · B Bolzano–Weierstrass theorem Every bounded sequence has a convergent subsequence.
Theorem · B Cauchy criterion A sequence of real numbers converges if and only if it is Cauchy.

1.3 Limits and continuity with epsilon and delta

Theorem · B Sequential criterion for continuity $f$ is continuous at $a$ if and only if $x_n \to a$ always implies $f(x_n) \to f(a)$ .
Theorem · B Intermediate value theorem If $f$ is continuous on $[a, b]$ and $f(a) < 0 < f(b)$ , then $f(c) = 0$ for some $c$ in $(a, b)$ .
Theorem · B Extreme value theorem A continuous function on a closed bounded interval $[a, b]$ attains a maximum and a minimum.
Theorem · B Continuous on a closed interval implies uniformly continuous A continuous function on $[a, b]$ is uniformly continuous there.

1.4 The derivative as best linear approximation

Theorem · B Differentiable implies continuous If $f$ is differentiable at $a$ , it is continuous at $a$ .
Theorem · B Chain rule If $g$ is differentiable at $a$ and $f$ at $g(a)$ , then $(f \circ g)'(a) = f'(g(a))\,g'(a)$ .
Theorem · B Interior extremum theorem If f has a maximum or minimum at an interior point c and is differentiable there, then f'(c) = 0.
Theorem · B Mean value theorem If f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) has f'(c) = (f(b) − f(a))/(b − a).
Corollary · B Zero derivative means constant If $f'(x) = 0$ for every x in an interval, f is constant there.
Theorem · B Taylor's theorem with Lagrange remainder If f has n + 1 derivatives, f(x) = T_n(x) + f^{(n+1)}(c)(x − a)^{n+1}/(n+1)! for some c between a and x.

1.5 The Riemann integral: area as a limit

Lemma · B Refinement lemma Adding cut points to a partition can only raise the lower sum and lower the upper sum.
Corollary · B Every lower sum is at most every upper sum For any two partitions $P$ and $Q$ of $[a, b]$ , $L(f, P) \le U(f, Q)$ .
Theorem · B Riemann–Darboux criterion
Theorem · B Darboux sums are integrals of step functions The sup of the lower sums equals the sup of the integrals of step functions below $f$ , and likewise for upper sums, so Apostol's integral and the Darboux integral agree.
Theorem · B Continuous functions are integrable If $f$ is continuous on $[a, b]$ , then $f$ is integrable on $[a, b]$ .
Theorem · B Linearity and monotonicity of the integral If $f, g$ are integrable and $c$ is a constant, then $f + g$ and $cf$ are integrable, with $\int (f+g) = \int f + \int g$ , $\int cf = c\int f$ , and $f \le g$ implies $\int f \le \int g$ .
Theorem · B Additivity of the integral over intervals If $a < c < b$ and $f$ is integrable on $[a, c]$ and on $[c, b]$ , then it is integrable on $[a, b]$ and $\int_a^b f = \int_a^c f + \int_c^b f$ .
Theorem · B Fundamental theorem of calculus, part 1 If $f$ is continuous on $[a, b]$ and $F(x) = \int_a^x f$ , then $F'(x) = f(x)$ for every $x$ .
Theorem · B Fundamental theorem of calculus, part 2 If $F$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $F' = f$ is integrable, then $\int_a^b f = F(b) - F(a)$ .

1.6 Series, power series, and uniform convergence

Theorem · B Geometric series $\sum_{k=0}^\infty r^k = \dfrac{1}{1 - r}$ when $|r| < 1$ , and the series diverges when $|r| \ge 1$ .
Theorem · B The nth-term test If $\sum a_k$ converges, then $a_k \to 0$ .
Theorem · B The harmonic series diverges $\sum_{k=1}^\infty \frac1k$ diverges, although $\frac1k \to 0$ .
Theorem · B Comparison test If $0 \le a_k \le b_k$ and $\sum b_k$ converges, then $\sum a_k$ converges; if $\sum a_k$ diverges, so does $\sum b_k$ .
Theorem · B Cauchy criterion for series
Theorem · B Absolute convergence implies convergence If $\sum |a_k|$ converges, then $\sum a_k$ converges, and $\left|\sum a_k\right| \le \sum |a_k|$ .
Theorem · B Ratio test
Theorem · B Alternating series test If $b_k$ decreases to $0$ , then $\sum (-1)^{k+1} b_k$ converges, and the error after $n$ terms is at most $b_{n+1}$ .
Theorem · B Rearranging an absolutely convergent series If $\sum a_k$ converges absolutely, every rearrangement converges to the same sum.
Theorem · B Uniform limit of continuous functions is continuous If each $f_n$ is continuous and $f_n \to f$ uniformly, then $f$ is continuous.
Theorem · B Weierstrass M-test If $|f_k(x)| \le M_k$ for all $x \in D$ and $\sum M_k$ converges, then $\sum f_k$ converges uniformly (and absolutely) on $D$ .
Lemma · B Convergence of power series (Abel's lemma) If $\sum c_k x_0^k$ converges, then $\sum c_k x^k$ converges absolutely for every $|x| < |x_0|$ .
Theorem · B Ratio formula for the radius If $c_k \ne 0$ and $\left|\frac{c_{k+1}}{c_k}\right| \to L$ , then $R = \frac1L$ (with $R = \infty$ if $L = 0$ and $R = 0$ if $L = \infty$ ).
Theorem · B Power series converge uniformly inside the radius If $0 < r < R$ , then $\sum c_k x^k$ converges uniformly on $[-r, r]$ , so its sum is continuous on $(-R, R)$ .
Theorem · B Integrating a uniform limit If continuous $f_n \to f$ uniformly on $[a, b]$ , then $\int_a^b f_n \to \int_a^b f$ .
Theorem · B Differentiating a uniform limit of derivatives If $f_n$ have continuous derivatives, $f_n' \to g$ uniformly and $f_n(a)$ converges, then $f_n \to f$ with $f' = g$ .
Theorem · B Properties of the exponential series $E' = E$ , $S' = C$ , $C' = -S$ , and $E(x + y) = E(x)\,E(y)$ , so $E(x) = e^x$ with $e = E(1) = 2.71828\ldots$
Theorem · B Product of absolutely convergent series If $\sum a_j$ and $\sum b_k$ converge absolutely, then $\sum_n c_n$ with $c_n = \sum_{j+k=n} a_j b_k$ converges to $\left(\sum a_j\right)\left(\sum b_k\right)$ .
Lemma · B The series C and S are cosine and sine For every real $x$ , $C(x) = \cos x$ and $S(x) = \sin x$ , where $\cos$ and $\sin$ are the unit-circle functions.
Theorem · B Euler's formula via power series $e^{i\theta} = \cos\theta + i\sin\theta$ for every real $\theta$ , where $e^{z} = \sum z^k/k!$ .

Terms this unit defines

Unit review

A mixed set of 12 quick checks drawn at random from the 56 in this unit's chapters. Each one says which chapter it comes from, so a miss tells you what to reread. Press "New set" for another mix.

Where it leads

Units that build on this one: Unit 4, Space and closeness, Unit 5, Differential calculus in many variables, Unit 8, Fourier analysis, Unit 9, Complex analysis, Unit 10, Probability and statistics, Unit 15, Optimisation, dynamics and computation.