the emants of real analysis, second edition (1976)

This is the book that turns calculus from a bag of tricks into a subject with reasons. Every theorem you half-remember from first-year calculus—the Intermediate Value Theorem, the Mean Value Theorem, the Fundamental Theorem of Calculus—gets a real proof here, built on completeness. Read it if you're a math major heading into analysis, a teacher who wants to know why the rules hold, or a self-learner who refuses to take 'it can be shown' for an answer.

1. 1Chapter 1

   Eleven axioms and one that matters most

   You open the book and meet the real number system R as a field with addition and multiplication satisfying 11 axioms. Then Bartle adds the one axiom that changes everything—completeness, the guarantee that every nonempty set bounded above has a least upper bound (supremum). From it he proves the Archimedean Property: if x > 0 and y is any real number, there exists a natural number n such that nx > y. No matter how small x, enough copies of it outrun any y. He immediately cashes this in for a second result: between any two real numbers there exists a rational number, using the Archimedean property to find a denominator n with 1/n < b - a.

   The idea

   Completeness is the load-bearing wall of analysis. The eleven field and order axioms describe arithmetic and size, but they hold equally for the rationals, where sequences like the decimal expansion of √2 have no limit. The supremum axiom is what makes limits exist at all, and the Archimedean Property is its first dividend: it converts 'arbitrarily small' into a concrete denominator you can hunt for. When you later prove that limits are unique or that integrals exist, you are quietly leaning on this chapter.

   Try this: Pick two real numbers, say a = 1.4142 and b = 1.4143, and actually find a rational between them using the Archimedean trick: choose n with 1/n < b - a, then locate an integer m with a < m/n < b.

2. 2Chapter 2

   Monotone, bounded, done: your first convergence theorem

   Chapter 2 hands you the cheapest convergence test in the book. A monotone increasing sequence that is bounded above converges to its supremum (and a bounded decreasing sequence converges to its infimum). Then the stakes rise. The chapter introduces subsequences and proves the Bolzano-Weierstrass Theorem via the nested interval property: every bounded sequence in R, no matter how chaotic, hides a convergent subsequence inside it. Finally comes the Cauchy Criterion for convergence: a sequence is convergent if and only if it is a Cauchy sequence, using the completeness of the reals.

   The idea

   These three results form a toolkit you'll use for the rest of the book. The Monotone Convergence Theorem lets you prove convergence without knowing the limit—just bound the sequence and let the supremum do the work. Bolzano-Weierstrass is the extraction principle: from any bounded mess, pull out a well-behaved thread. The Cauchy Criterion matters most when the limit is unknown or lives in a space you can't see directly; it tests convergence using only the terms themselves. All three are completeness wearing different clothes.

   Try this: Take the sequence a₁ = 1, a_{n+1} = 1 + 1/aₙ. Show it's monotone after the first term or two, bound it above by 2, and conclude it converges—without ever computing its limit.

3. 3Chapter 3

   ε-δ: the definition that ends the guessing

   Bartle puts the ε-δ definition of a limit on the table: f has limit L at c if for every ε > 0 you can find δ > 0 so that x near c forces f(x) near L. The example f(x) = (x^2 - 1)/(x - 1) for x ≠ 1 has limit 2 as x approaches 1 despite being undefined at 1.

   The idea

   The ε-δ definition reverses the power dynamic of a calculus course. Instead of 'gets close to,' which means whatever the reader wants, it demands a quantified challenge game: your opponent picks ε, you answer with δ. If you can always answer, the limit exists. This is why removable discontinuities and holes don't block limits, and why the same machinery later defines continuity by simply requiring the limit to equal f(c).

   Try this: Prove lim_{x→2} (3x + 1) = 7 with the definition: given ε, exhibit δ = ε/3 and verify the implication |x - 2| < δ ⟹ |f(x) - 7| < ε.

4. 4Chapter 4

   Continuity, assembled from limits

   Continuity gets its precise form: f is continuous at c if for every ε > 0 there exists δ > 0 such that if |x - c| < δ, then |f(x) - f(c)| < ε. It is proved that sums, products, and quotients (where denominator non-zero) of continuous functions are continuous. The payoff arrives in three theorems. The Intermediate Value Theorem is stated: if f is continuous on [a,b] and k is between f(a) and f(b), then there exists c in [a,b] such that f(c) = k, proved using the supremum of the set {x : f(x) ≤ k}. The Boundedness Theorem is proved: a function continuous on a closed bounded interval [a,b] is bounded on that interval. The Maximum-Minimum (Extreme Value) Theorem is proved: a continuous function on a closed bounded interval attains its maximum and minimum values, using the Bolzano-Weierstrass theorem.

   The idea

   Closed and bounded is the magic phrase in this chapter. The proofs show why completeness earns its keep—the Intermediate Value Theorem conjures the point c out of a supremum, and the Extreme Value Theorem harvests a convergent subsequence from Bolzano-Weierstrass. Continuity alone is a local condition; it's the compactness of [a,b] that turns it into global control.

   Try this: Show f(x) = 1/x on (0, 1] is continuous but unbounded, then explain precisely which hypothesis of the Boundedness Theorem fails.

5. 5Chapter 5

   The derivative, defined honestly

   The derivative of a function f at a point c is defined as f'(c) = lim_{h->0} (f(c+h) - f(c))/h, if the limit exists, and it is proved that differentiability implies continuity. Then the chapter's centerpiece. Rolle's Theorem is proved: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) such that f'(c)=0. The Mean Value Theorem is then proved: if f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) such that f'(c) = (f(b)-f(a))/(b-a), using Rolle's theorem. L'Hospital's Rule (for the 0/0 indeterminate form) is stated and proved using the Cauchy Mean Value Theorem, with the example lim_{x→0} (sin x)/x = 1, though the proof uses the derivative of sin at 0.

   The idea

   The Mean Value Theorem is the bridge between local and global: it converts information about one derivative at one unknown point into a statement about average change over a whole interval. Nearly every derivative fact you use—monotonicity tests, error bounds for approximations—is the MVT in disguise. L'Hospital's Rule is not a definition but a theorem with hypotheses, and its proof via the Cauchy Mean Value Theorem shows why the rule needs differentiability near the point, not just at it.

   Try this: Use the Mean Value Theorem to prove |sin a - sin b| ≤ |a - b| for all real a, b, identifying the c whose existence the theorem guarantees.

6. 6Chapter 6

   The Riemann integral, from partitions up

   Chapter 6 builds the integral from bare parts. A partition P of [a,b] is a finite set of points {x_0, x_1, ..., x_n} with a = x_0 < x_1 < ... < x_n = b; its norm is ||P|| = max{x_i - x_{i-1}}; a Riemann sum is S(f;P) = Σ f(t_i)(x_i - x_{i-1}) for tags t_i in [x_{i-1}, x_i]; and f is Riemann integrable if there exists L such that for every ε > 0 there exists δ > 0 such that for any partition P with ||P|| < δ and any tags, |S(f;P) - L| < ε. It is proved that a continuous function on [a,b] is Riemann integrable. Then the Fundamental Theorem of Calculus is proved in both parts: if F is differentiable on [a,b] and F' = f is integrable on [a,b], then ∫_a^b f = F(b) - F(a); and if f is continuous on [a,b], then F(x) = ∫_a^x f is differentiable on [a,b] and F'(x) = f(x).

   The idea

   The tag-based definition is sturdier than the one you saw in calculus: it demands that ALL Riemann sums with fine enough partitions land near L, no matter where the tags fall. That's why the theorem 'continuous implies integrable' is real content—the tags could conspire to sample the worst points, and continuity is exactly what stops them. The two parts of the Fundamental Theorem point in opposite directions: one evaluates integrals via antiderivatives, the other manufactures derivatives from integrals. Together they say differentiation and integration are inverse operations, under hypotheses worth checking.

   Try this: Compute a Riemann sum for f(x) = x² on [0,1] using the partition {0, 1/4, 1/2, 1} with tags at the left endpoints, and compare it to the exact value 1/3.

7. 7Chapter 7

   When limits of functions betray you

   Here the book shows you a failure. The example f_n(x) = x^n on [0,1] is given, which converges pointwise to 0 for x < 1 and to 1 at x = 1, but not uniformly. The repair is uniform convergence, defined as: (f_n) converges uniformly to f on A if for every ε > 0 there exists N such that for all n ≥ N and all x in A, |f_n(x) - f(x)| < ε. It is proved that the uniform limit of a sequence of continuous functions is continuous, and that uniform convergence allows interchange of limit and integral. For series, the Weierstrass M-Test for uniform convergence of series is stated: if |f_n(x)| ≤ M_n for all x in A and Σ M_n converges, then Σ f_n converges uniformly on A. The example of the series Σ x^n/n^2 on [0,1] is given, which converges uniformly by the M-test with M_n = 1/n^2, since Σ 1/n^2 converges.

   The idea

   Pointwise convergence lets each point converge on its own schedule; uniform convergence forces a single schedule for everyone. That shared schedule is what lets you interchange limits with continuity and integration—the ε that works at one point works everywhere. The M-Test is the workhorse because it reduces a question about functions to a question about numbers, a series you can check with the comparison test. Keep the xⁿ example in mind: it's the canonical reminder that a limit of nice objects need not be nice.

   Try this: For f_n(x) = xⁿ on [0, 1/2], find an explicit N(ε) that works for every x at once, and confirm the convergence is uniform there even though it fails on [0,1].

8. 8Chapter 8

   Power series and their radius of trust

   The final chapter treats infinite polynomials. The radius of convergence R is defined using the Cauchy-Hadamard formula: 1/R = lim sup |a_n|^{1/n}. It is proved that a power series can be differentiated term-by-term within its interval of convergence, and the radius of convergence of the derivative is the same.

   The idea

   Term-by-term differentiation is the result that justifies treating power series like polynomials, and it's far from obvious: general series of functions cannot be differentiated term-by-term without extra hypotheses. The Cauchy-Hadamard formula is worth pausing over because it uses lim sup, the tool from Chapter 2—when ordinary limits of |a_n|^{1/n} may not exist, the lim sup still pins down R exactly. The equal radii of a series and its derivative also explain why functions like e^x have series that converge everywhere: differentiation never shrinks the domain of trust.

   Try this: Apply the Cauchy-Hadamard formula to Σ xⁿ/n! by computing lim sup (1/n!)^{1/n} = 0, and conclude R = ∞; then differentiate the series term-by-term to recover the same series.

Math majors meeting rigorous analysis for the first time, and self-learners who want calculus's theorems proved rather than asserted. Teachers of introductory analysis will also find the ε-δ scaffolding unusually clean.

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