30:T4a0,We collect theorems (or properties, if you prefer) about the real numbers that are routinely used in calculus and analysis. The point is that it does not matter which one or another *construction* of the real numbers is under the hood. In other words, we take the real numbers as any *ordered field* (field with a total order ${tex`\leq`}) containing the rational numbers that satisfies the **axiom of completeness**. Many, if not all, of the following may be taken as the axiom of completeness — and the rest would be theorems — and each one is saying in its own way that *the real line does not have any gaps*; for more, see * James Propp, [Real Analysis in Reverse](https://arxiv.org/abs/1204.4483), *The American Mathematical Monthly* (2013) Note that each of the following theorems or properties of real numbers is *second-order*, in that it refers to *all* instances of a "higher-level construct" of a certain type (non-empty **sets**, bounded **sequences**, or continuous **functions**, of real numbers) as opposed to just the individual numbers. While reading each statement, think about what happens if the real numbers had a gap. Can you come up with a counterexample?31:T4c5,const_fn = function({f="f", x="x", a="a", b="b", c="c", color="red"}) { f = (f != "f") ? "\\color{" + color + "}" + f + "\\color{black}" : "f"; const statement = md` **Theorem**: If ${tex`${f}`} is continuous on an interval of the real line ${tex`\mathbb R`} and differentiable, with ${tex`${f}'(x)=0`} on the interior, then ${tex`${f}`} is a constant function. ` const proof1 = md` *Proof*: Take any two points ${tex`${a}`} and ${tex`${b}`} of the interval. By the ${MVT({f:f}).name()}, there exists a point ${tex`c\in (a,b)`} such that ${MVT({f:f}).formula} By the hypothesis, ${tex`${f}'(c)=0`}, which implies that ${tex`${f}(b)=${f}(a)`}. Since ${tex`a`} and ${tex`b`} are arbitrary, ${tex`${f}`} must be a constant function. ` const proof2 = md` *Proof*: Take any two points ${tex`${a}`} and ${tex`${b}`} of the interval. By the ${MVI({f:f}).name()}, ${MVI({f:f, M:0}).formula} hence ${tex`${f}(b)=${f}(a)`}. Since ${tex`a`} and ${tex`b`} are arbitrary, ${tex`${f}`} must be a constant function. ` const thm = new Proposition("Constant Function Theorem", statement, [proof1, proof2]) thm.name_link = "[Constant Function Theorem](https://observablehq.com/@neobourbaki/real-numbers)" return thm }32:T736,differentiable = function({f="f", I="I", x="x", x_0="x_0", d="d", h="h"}) { const def1 = md` **Definition**: A function ${tex`${f}:${I}\to\mathbb R`} ${real_number().hidden("")} on an open interval ${tex`${I}\subseteq\mathbb R`} is **differentiable at** ${tex`${x_0}\in ${I}`} if there exists ${tex`${d}\in\mathbb R`} such that ${tex.block`${f}(${x_0}+${h}) = ${f}(${x_0}) + ${d}${h} + o(${h})`} as ${tex`${h}\to 0`}, or in other words, ${tex.block`\lim_{${h}\to 0} \frac{${f}(${x_0}+${h})-${f}(${x_0})}{${h}}`} ${continuous({}).hidden("exists")}. In that case, ${tex`${d}`} is called the **derivative of** ${tex`${f}`} **at** ${tex`${x}`}, and is denoted ${tex`${f}'(${x})`}. If ${tex`${f}`} is differentiable at all ${tex`${x}\in ${I}`}, we say that ${tex`${f}`} is **differentiable**, and the values ${tex`${f}'(${x})`} indeed define a function ${tex`${f}': ${I}\to\mathbb R`}, called the **derivative** of ${tex`${f}`}.` const def2 = md` **Definition**: A function ${tex`${f}:${I}\to\mathbb R`} ${real_number().hidden("")} on an open interval ${tex`${I}\subseteq\mathbb R`} is **differentiable at** ${tex`${x_0}\in ${I}`} if there exists a function ${tex`g: ${I}\to\mathbb R`} such that ${tex.block`${f}(${x})=${f}(${x_0})+g(${x})(${x}-${x_0})`} and that ${tex`g(${x})`} is ${continuous({}).name()} at ${tex`${x_0}`}. In that case, ${tex`g(x_0)`} is called the **derivative of** ${tex`${f}`} **at** ${tex`${x_0}`}, and is also denoted ${tex`${f}'(${x_0})`}. If ${tex`${f}`} is differentiable at all ${tex`${x}\in ${I}`}, we say that ${tex`${f}`} is **differentiable**, and the function ${tex`${x}\mapsto ${f}'(${x})`} is called the **derivative** of ${tex`${f}`}.` const def = new Definition("differentiable", [def1, def2]) def.name_link = (s="differentiable")=> "[" + s + "](./real-numbers#differentiable_tag)" return def }33:[] 2e:{"title":"Real numbers","cells":[{"id":0,"value":"# Real numbers\n\nThe notion of **real numbers** has been in existence for a long time without a solid foundation. For much of its history it is better known by the name of the **continuum**, implicit in Euclidean geometry as *all* the points on a straight line. As numbers, they could simply be regarded as decimal expressions that go on forever, not repeating itself or following any pattern whatsoever.\n\nDespite the following basic fact, which implies that there are way more real numbers than what one would encounter \"in nature\" (rational, algebaric, or computable numbers), the *totality* of real numbers greatly *simplifies* the mathematics of the continuous, which incidentally also has remarkable applications for the discrete (e.g., analytic number theory).","pinned":false,"hidden":false,"mode":"md"},{"id":46,"value":"R_uncountable().statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":50,"value":"$30","pinned":false,"hidden":false,"mode":"md"},{"id":139,"value":"## 1. Functions of real numbers","pinned":false,"hidden":false,"mode":"md"},{"id":180,"value":"differentiable_tag = md`### a) differentiability`","pinned":false,"hidden":false,"mode":"ojs"},{"id":127,"value":"const_fn({}).statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":145,"value":"const_fn({}).proof[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":771,"value":"MVI({}).statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":130,"value":"MVT({}).statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":288,"value":"MVT({}).proof","pinned":false,"hidden":false,"mode":"ojs"},{"id":356,"value":"differentiable({}).definition[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":169,"value":"continuous_tag = md`### b) continuity`","pinned":false,"hidden":false,"mode":"ojs"},{"id":186,"value":"IVT({}).statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":259,"value":"IVT({}).proof({})","pinned":false,"hidden":false,"mode":"ojs"},{"id":193,"value":"EVT({}).statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":176,"value":"continuous({}).definition","pinned":false,"hidden":false,"mode":"ojs"},{"id":136,"value":"## 2. Sequences of real numbers","pinned":false,"hidden":false,"mode":"md"},{"id":20,"value":"Cauchy_sequence({}).statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":400,"value":"uniqueness_of_limit().statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":226,"value":"monotone_convergence({}).statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":93,"value":"convergence({}).definition","pinned":false,"hidden":false,"mode":"ojs"},{"id":33,"value":"Bolzano_Weierstrass().statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":134,"value":"## 3. Sets of real numbers","pinned":false,"hidden":false,"mode":"md"},{"id":6,"value":"least_upper_bound({}).statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":388,"value":"uniqueness_of_lub().statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":81,"value":"cut_property({}).statement","pinned":false,"hidden":false,"mode":"ojs"},{"id":485,"value":"## 0. Constructions of real numbers","pinned":false,"hidden":false,"mode":"md"},{"id":427,"value":"real_number().definition","pinned":false,"hidden":false,"mode":"ojs"},{"id":600,"value":"computable_number().definition","pinned":false,"hidden":false,"mode":"ojs"},{"id":43,"value":"R_uncountable = function() {\n const statement = md`\n**Theorem** (Cantor): The set ${tex`\\mathbb R`} of real numbers is uncountable.\n`\n const proof = md`\n*Proof:* \n`\n return new Proposition(\"uncountability of the reals\", statement)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":122,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":769,"value":"MVI = function({f=\"f\", x=\"x\", a=\"a\", b=\"b\", c=\"c\", M=\"M\", color=\"red\"}) {\n f = (f != \"f\") ? \"\\\\color{\" + color + \"}\" + f + \"\\\\color{black}\" : \"f\";\n const formula = tex.block`${f}(${b}) - ${f}(${a}) \\leq ${M} (${b}-${a})`\n \n const statement = md`\n**Mean Value Inequality**: If ${tex`${f}`} is continuous on the interval ${tex`[${a},${b}]`} and differentiable on ${tex`(${a},${b})`}, and that ${tex`${f}'(x)\\leq ${M}`} for some ${tex`${M}`}, then \n${formula}\n`\n const proof = md`*Proof:*`\n const thm = new Proposition(\"mean value inequality\", statement, proof)\n thm.formula = formula\n return thm\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":116,"value":"MVT = function({f=\"f\", x=\"x\", a=\"a\", b=\"b\", c=\"c\", color=\"red\"}) {\n f = (f != \"f\") ? \"\\\\color{\" + color + \"}\" + f + \"\\\\color{black}\" : \"f\";\n const formula = tex.block`${f}'(${c}) = \\frac{${f}(${b})-${f}(${a})}{${b}-${a}}`\n \n const statement = md`\n**Mean Value Theorem** (Lagrange): If ${tex`${f}`} is continuous on the interval ${tex`[${a},${b}]`} and differentiable on ${tex`(${a},${b})`}, then there exists a point ${tex`${c}\\in (${a},${b})`} such that\n${formula}\n`\n const proof = md`*Proof:* It follows from the special case (known as Rolle's theorem) that ${tex`${f}(a)=${f}(b)`}, which in turn is a consequence of the ${EVT({}).name()} and the definition of ${differentiable({}).hidden(\"differentiability\")}.`\n const thm = new Proposition(\"mean value theorem\", statement, proof)\n thm.formula = formula\n return thm\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":351,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":184,"value":"IVT = function({f=\"f\", x=\"x\", a=\"a\", b=\"b\", c=\"c\", color=\"red\"}) {\n f = (f != \"f\") ? \"\\\\color{\" + color + \"}\" + f + \"\\\\color{black}\" : \"f\";\n const statement = md`\n**Intermediate Value Theorem**: If ${tex`${f}`} is continuous on the interval ${tex`[${a},${b}]`} and ${tex`${f}(${a})<${f}(${b})`}, and ${tex`y`} is any intermediate value, i.e., ${tex`f(a) md`\n*Proof:* Let ${tex`${S}`} be the subset of ${tex`I`} for which ${tex`${f}(${x})<${y}`}. Then ${tex`${S}`} is nonempty because ${tex`a\\in S`}, and is bounded from above by ${tex`${b}`}. Then the least upper bound ${tex`c`} of ${tex`S`} exists. We claim that ${tex`c\\in I`} and ${tex`f(c) = y`}.`\n return new Proposition(\"intermediate value theorem\", statement, proof)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":191,"value":"EVT = function({f=\"f\", x=\"x\", a=\"a\", b=\"b\", c=\"c\", color=\"red\"}) {\n f = (f != \"f\") ? \"\\\\color{\" + color + \"}\" + f + \"\\\\color{black}\" : \"f\";\n var statement = md`\n**Extreme Value Theorem**: If ${tex`${f}`} is continuous on the interval ${tex`[${a},${b}]`}, then ${tex`${f}`} attains its maximum, i.e., there exists ${tex`${c}\\in [${a},${b}]`} such that ${tex`${f}(${x})\\leq ${f}({${c}})`} for all ${tex`${x}\\in [${a}, ${b}]`}.\n`\n var proof = md`${continuous({}).name()}, and ${least_upper_bound({}).name}`\n return new Proposition(\"extreme value theorem\", statement, proof)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":173,"value":"continuous = function({f=\"f\", x=\"x\", x_0=\"x_0\"}) {\n const definition = md`

\n**Definition**: A ${real_number().hidden(\"\")} function ${tex`${f}:I\\to\\mathbb R`} on an interval ${tex`I\\subseteq\\mathbb R`} is **continuous at** ${tex`${x_0}\\in I`} if for any ${tex`\\epsilon>0`}, there exists ${tex`\\delta`} such that \n${tex.block`|${f}(${x})-${f}(${x_0})|<\\epsilon`}\nfor all ${tex`${x_0}\\in I`} with ${tex`|${x_0}-${x}|<\\delta`}. We say ${tex`f`} is **continuous** if it is continuous at each ${tex`x\\in I`}.\n`\n const def = new Definition(\"continuous\", definition)\n def.name_link = (s=\"continuous\") => \"[\" + s + \"](./real-numbers#continuous_tag)\"\n return def\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":18,"value":"Cauchy_sequence = function({x=\"x\", N=\"N\"}) {\n const statement = md`\n**Theorem** (Cauchy): Suppose that ${tex`\\{${x}_n\\}`} is a Cauchy sequence, i.e., for any ${tex`\\epsilon>0`}, there exists ${tex`${N}`} such that\n${tex.block`|${x}_m-${x}_n|<\\epsilon \\quad\\forall\\,m,n > ${N}`}\nthen there exists ${tex`${x}\\in\\mathbb R`} such that ${tex`${x}_n\\to ${x}`}.\n`\n return new Proposition(\"completeness of reals\", statement)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":394,"value":"uniqueness_of_limit = function(){\n const statement = md`\n**Proposition**: The limit of a sequence ${tex`\\{x_n\\}`} is unique.\n`\n return new Proposition(\"uniquess of limit\", statement)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":224,"value":"monotone_convergence = function({x=\"x\", n=\"n\"}) {\n const x_n = x + \"_\" + \"n\"\n const statement = md`\n**Theorem**: If ${tex`\\{${x_n}\\}`} is a bounded monotonically increasing sequence, i.e.,\n1. there exists ${tex`M`} such that ${tex`${x_n} \\leq M`} for all ${tex`${n}`}\n2. ${tex`${x}_{${n}+1} \\geq ${x_n}`} for all ${tex`${n}`}\n\nthen the sequence ${tex`\\{${x_n}\\}`} converges.\n`\n const proof = md`\n*Proof:* Follows from the least upper bound property.\n`\n return new Proposition(\"monotone convergence\", statement, proof)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":90,"value":"convergence = function({x=\"x\", n=\"n\", L=\"x\"}) {\n const x_n = x + \"_\" + n\n const def = md`\n**Definition**: A sequence ${tex`\\{${x_n}\\}`} of real numbers **converges** (in ${tex`\\mathbb R`}) if there exists ${tex`${L}\\in\\mathbb R`} such that for all ${tex`\\epsilon>0`}, there exists ${tex`N\\in\\mathbb N`} such that \n${tex.block`|${x_n}-${L}|<\\epsilon \\quad\\forall\\,n > N`}\nIn such case, we also say that the sequence ${tex`\\{${x_n}\\}`} **converges to** ${tex`${L}`} as ${tex`${n}`} tends to infinity, or that the **limit** of ${tex`${x_n}`} is ${tex`${L}`}, denoted\n${tex.block`\\lim_{n\\to\\infty} x_n = ${L}`}\nor \n${tex.block`${x_n}\\to ${L}\\quad\\text{as}\\quad {n}\\to\\infty`}\n`\n return new Definition(\"convergence\", def)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":28,"value":"Bolzano_Weierstrass = function() {\n const statement = md`\n**Theorem** (Bolzano-Weierstrass): Suppose that ${tex`\\{x_n\\}`} is a bounded sequence, then there exists a subsequence, i.e., ${tex`\\{x_{n_j}\\}`} for some order-preserving map ${tex`j\\mapsto n_j`} of ${tex`\\mathbb N\\to\\mathbb N`}, that converges.\n` \n return new Proposition(\"Bolzano-Weierstrass theorem\", statement)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":4,"value":"least_upper_bound = function({S=\"S\", c=\"c\"}) {\n const statement = md`\n**Theorem** (Dedekind): If ${tex`S`} is a non-empty subset of ${tex`\\mathbb R`} that is bounded from above, then there exists a **least upper bound** (or **supremum**) of ${tex`S`}, i.e., a real number ${tex`c`} such that\n1. ${tex`x\\leq c`} for all ${tex`x\\in S`}, and\n2. If ${tex`d`} is any real number such that ${tex`x\\leq d`} for all ${tex`x\\in S`}, then ${tex`c\\leq d`}.\n`\n const proof = md`*Proof:* Follows from the definition of ${real_number().name}.`\n return new Proposition(\"least upper bound property\", statement)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":383,"value":"uniqueness_of_lub = function(){\n const statement = md`\n**Proposition**: The least upper bound of a set ${tex`S\\subseteq\\mathbb R`} is unique.\n`\n const proof = md`\n*Proof:* Suppose ${tex`c_1`} and ${tex`c_2`} are both least upper bounds of ${tex`S`}. Then ${tex`c_1\\leq c_2`} and ${tex`c_2\\leq c_1`}, hence ${tex`c_1 = c_2`}.\n`\n return new Proposition(\"uninqueness of least upper bound\", statement, proof)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":79,"value":"cut_property = function({A=\"A\", B=\"B\", c=\"c\"}) {\n const statement = md`\n**Theorem** (Dedekind): If ${tex`${A}`} and ${tex`${B}`} are disjoint subsets of ${tex`\\mathbb R`} whose union is all of ${tex`\\mathbb R`}, and every element of ${tex`${A}`} is less than every element of ${tex`${B}`}, then there exists ${tex`${c}\\in\\mathbb R`} such that every number in ${tex`${A}`} is less than or equal to ${tex`${c}`} and every number in ${tex`${B}`} is greater than or equal to ${tex`${c}`}.\n`\n return new Proposition(\"Dedekind cut property\", statement)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":421,"value":"real_number = function(){\n const def = md`\n**Definition** (Dedekind): A **real number** is a cut of the set ${tex`\\mathbb Q`} of rational numbers, i.e., a partition ${tex`A\\coprod B`} of ${tex`\\mathbb Q`} such that ${tex`a0}\\to\\mathbb Q\\times\\mathbb Q`}, denoted ${tex`x\\mapsto (a_x,b_x)`} such that\n\n1. ${tex`0\\leq b_x-a_x \\leq x`}\n2. ${tex`a_x \\leq a_y`} and ${tex`b_y\\leq b_x`} for any ${tex`y