https://teddyrookbook.com/math/real-analysis/Recent content in Real Analysis on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/real-analysis/absolute-value/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/absolute-value/|x − a| < ε means x is within ε of a; |a + b| ≤ |a| + |b| and |a| − |b| ≤ |a − b| turn distances into bounds. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/sup-inf/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/sup-inf/The supremum of a set is its least upper bound and the infimum its greatest lower bound; neither has to belong to the set. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/approximation-property/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/approximation-property/An upper bound s is the supremum exactly when, for every ε > 0, some element of the set is bigger than s − ε. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/completeness/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/completeness/Every nonempty set of reals that is bounded above has a supremum; from it, for every ε > 0 some whole number n has 1/n < ε. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/density/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/density/Between any two real numbers a < b lies a rational: pick n with 1/n < b − a, and some multiple m/n lands strictly between them. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/countable-sets/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/countable-sets/A set is countable when its elements can be listed a₁, a₂, a₃, …; the rationals are countable, but Cantor's diagonal argument shows the reals are not. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/sequence-limit/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/sequence-limit/aₙ → L when for every ε > 0 there is an N with |aₙ − L| < ε for all n ≥ N; for a given ε you can find such an N and check it. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/epsilon-n-proofs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/epsilon-n-proofs/Work backward on scratch paper to find N in terms of ε; then write the proof forward: let ε > 0, choose N, take n ≥ N and show |aₙ − L| < ε. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/divergence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/divergence/Every convergent sequence is bounded; to show aₙ does not converge to L, find one ε > 0 for which no N works. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/limit-laws/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/limit-laws/If aₙ → a and bₙ → b, then aₙ + bₙ → a + b and aₙbₙ → ab; the proofs split ε between the two pieces and use that convergent sequences are bounded. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/squeeze-order/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/squeeze-order/Limits keep ≤: if aₙ ≤ bₙ then lim aₙ ≤ lim bₙ; and a sequence trapped between two sequences with the same limit has that limit too. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/monotone-convergence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/monotone-convergence/A bounded monotone sequence converges, to the supremum of its terms if it increases; for aₙ₊₁ = g(aₙ), show monotone and bounded, then solve L = g(L). Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/subsequences/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/subsequences/Every subsequence of a convergent sequence has the same limit, so two subsequences with different limits prove that a sequence diverges. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/bolzano-weierstrass/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/bolzano-weierstrass/Every bounded sequence has a convergent subsequence: halve the interval again and again, each time keeping a half that holds infinitely many terms. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/limsup-liminf/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/limsup-liminf/lim sup aₙ is the limit of the tail suprema sup{aₖ : k ≥ n}; it is the largest subsequential limit, and aₙ converges exactly when lim sup aₙ = lim inf aₙ. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/cauchy-sequences/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/cauchy-sequences/aₙ is Cauchy when for every ε > 0 some N makes |aₘ − aₙ| < ε for all m, n ≥ N; in ℝ a sequence is Cauchy exactly when it converges. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/series-cauchy/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/series-cauchy/∑aₖ converges when its partial sums do; by the Cauchy criterion, blocks aₘ₊₁ + ⋯ + aₙ far out must be small, and in particular aₖ → 0. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/absolute-convergence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/absolute-convergence/If |aₖ| ≤ bₖ and ∑bₖ converges, the Cauchy criterion makes ∑aₖ converge; in particular, absolute convergence implies convergence. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/condensation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/condensation/For a decreasing sequence of positive terms, ∑aₙ and ∑2ᵏa₂ₖ converge or diverge together; it settles the p-series and ∑1/(n(ln n)ᵖ). Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/root-ratio-limsup/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/root-ratio-limsup/If lim sup |aₖ|^(1/k) < 1 the series converges absolutely, and if it is > 1 the series diverges; the root test works even when the ratios jump around. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/alternating-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/alternating-series/If bₖ decreases to 0, ∑(−1)ᵏ⁺¹bₖ converges, and the error after n terms is less than bₙ₊₁: the partial sums zigzag and trap the sum. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/rearrangements/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/rearrangements/Rearranging an absolutely convergent series keeps its sum; a conditionally convergent series can be rearranged to add up to any number you like (Riemann). Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/function-limits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/function-limits/lim f(x) = L as x → a when for every ε > 0 some δ > 0 gives |f(x) − L| < ε whenever 0 < |x − a| < δ; any smaller positive δ works too. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/continuity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/continuity/f is continuous at a when for every ε > 0 some δ > 0 gives |f(x) − f(a)| < ε whenever |x − a| < δ; for x² the classic choice is δ = min(1, ε/(2|a| + 1)). Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/sequential-criterion/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/sequential-criterion/f is continuous at a exactly when f(xₙ) → f(a) for every sequence xₙ → a; two sequences giving different limits prove a discontinuity. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/ivt/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/ivt/A continuous function on [a, b] takes every value between f(a) and f(b); the proof uses c = sup{x : f(x) < 0}, and bisection traps c to any accuracy. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/evt/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/evt/A continuous function on a closed, bounded interval attains a maximum and a minimum; drop any hypothesis and the supremum may not be attained. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/uniform-continuity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/uniform-continuity/f is uniformly continuous on S when one δ works for every point of S at once; continuous functions on [a, b] are, but 1/x on (0, 1) is not. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/derivative-definition/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/derivative-definition/f′(a) = lim (f(a + h) − f(a))/h as h → 0; the definition handles what the rules can't, like x² sin(1/x) at 0. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/differentiability/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/differentiability/Differentiable at a implies continuous at a, but not conversely; at a break point the one-sided derivatives must exist and agree. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/mean-value-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/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); Rolle's theorem is the case f(a) = f(b). Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/mvt-inequalities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/mvt-inequalities/If |f′| ≤ K on an interval, then |f(x) − f(y)| ≤ K|x − y| there; this proves bounds like |sin x − sin y| ≤ |x − y| and gives δ = ε/K. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/taylor-remainder/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/taylor-remainder/f(x) = Pₙ(x) + f⁽ⁿ⁺¹⁾(c)(x − a)ⁿ⁺¹/(n + 1)! for some c between a and x; a bound on f⁽ⁿ⁺¹⁾ bounds the error. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/upper-lower-sums/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/upper-lower-sums/For a partition P, U(f, P) uses the supremum of f on each piece and L(f, P) the infimum; every lower sum is at most every upper sum, and refining P brings them together. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/integrability-criterion/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/integrability-criterion/f is integrable exactly when for every ε > 0 some partition has U(f, P) − L(f, P) < ε; for monotone f on n equal pieces the gap is |f(b) − f(a)|(b − a)/n. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/integrable-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/integrable-functions/Continuous functions, monotone functions and bounded functions with finitely many jumps are integrable; Dirichlet's function is not, since every U = b − a and every L = 0. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/integral-properties/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/integral-properties/If m ≤ f ≤ M on [a, b], then m(b − a) ≤ ∫f ≤ M(b − a); a continuous f equals its average value at some c in [a, b]. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/ftc/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/ftc/F(x) = ∫ₐˣ f is continuous, and F′(c) = f(c) wherever f is continuous; at a jump of f, F has a corner. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/riemann-sum-limits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/riemann-sum-limits/Because continuous functions are integrable, every Riemann sum tends to the integral; so a limit like (1/n)∑f(k/n) is ∫₀¹ f(x) dx. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/pointwise-convergence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/pointwise-convergence/fₙ → f pointwise when fₙ(x) → f(x) at each x separately; a pointwise limit of continuous functions can jump, as xⁿ on [0, 1] does. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/uniform-convergence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/uniform-convergence/fₙ → f uniformly when ‖fₙ − f‖ = sup |fₙ(x) − f(x)| → 0: one N works for every x at once. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/uniform-limits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/uniform-limits/A uniform limit of continuous functions is continuous, and ∫fₙ → ∫f; without uniform convergence, the limit of the integrals can differ from the integral of the limit. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/weierstrass-m-test/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/weierstrass-m-test/If |fₖ(x)| ≤ Mₖ for every x and ∑Mₖ converges, then ∑fₖ converges uniformly (and absolutely). Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/power-series-radius/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/power-series-radius/∑cₙxⁿ has radius R = 1/lim sup |cₙ|^(1/n); it converges uniformly on every [−r, r] with r < R, so its sum is continuous inside the interval. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/metric-spaces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/metric-spaces/A metric d measures distance: d(x, y) > 0 unless x = y, d(x, y) = d(y, x), and d(x, z) ≤ d(x, y) + d(y, z); the taxicab, Euclidean and max metrics on ℝ² are three examples. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/open-closed-sets/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/open-closed-sets/A set is open when every point has a ball around it inside the set, and closed when it contains all its limit points; the closure adds the limit points. Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/compactness/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/compactness/K is compact when every open cover has a finite subcover; in ℝⁿ that happens exactly when K is closed and bounded (Heine–Borel). Real Analysis · Math Ladderhttps://teddyrookbook.com/math/real-analysis/contraction-mapping/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/real-analysis/contraction-mapping/If d(g(x), g(y)) ≤ k·d(x, y) with k < 1 on a complete space, g has exactly one fixed point, and the nth iterate is within kⁿd(x₁, x₀)/(1 − k) of it. Real Analysis · Math Ladder