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单词 1133TheAlgebraicStructureOfCauchyReals
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11.3.3 The algebraic structure of Cauchy reals


We first define the additivePlanetmathPlanetmath structureMathworldPlanetmath (𝖼,0,+,-). Clearly, the additive unit element0 is just 𝗋𝖺𝗍(0), while the additive inverse -:𝖼𝖼 is obtained as theextensionPlanetmathPlanetmathPlanetmathPlanetmath of the additive inverse -:, using \\autorefRC-extend-Q-Lipschitzwith Lipschitz constant 1. We have to work a bit harder for additionPlanetmathPlanetmath.

Lemma 11.3.1.

Suppose f:Q×QQ satisfies, for all q,r,s:Q,

|f(q,s)-f(r,s)||q-r|  𝑎𝑛𝑑  |f(q,r)-f(q,s)||r-s|.

Then there is a function f¯:Rc×RcRc such thatf¯(rat(q),rat(r))=f(q,r) for all q,r:Q. Furthermore,for all u,v,w:Rc and q:Q+,

uϵvf¯(u,w)ϵf¯(v,w)𝑎𝑛𝑑vϵwf¯(u,v)ϵf¯(u,w).
Proof.

We use (𝖼,)-recursion to construct the curried form of f¯ as a map𝖼A where A is the space of non-expanding real-valuedfunctions:

A:\\setofh:𝖼𝖼|(ϵ:+).(u,v:𝖼).uϵvh(u)ϵh(v).

We shall also need a suitable ϵ on A, which we define as

(hϵk):(u:𝖼).h(u)ϵk(u).

Clearly, if (ϵ:+).hϵk then h(u)=k(u) for all u:𝖼, so is separated.

For the base case we define f¯(𝗋𝖺𝗍(q)):A, where q:, as theextension of the Lipschitz map λr.f(q,r) from to 𝖼𝖼, asconstructed in \\autorefRC-extend-Q-Lipschitz with Lipschitz constant 1. Next, for aCauchy approximation x, we define f¯(𝗅𝗂𝗆(x)):𝖼𝖼 as

f¯(𝗅𝗂𝗆(x))(v):𝗅𝗂𝗆(λϵ.f¯(xϵ)(v)).

For this to be a valid definition, λϵ.f¯(xϵ)(v) should be aCauchy approximation, so consider any δ,ϵ:. Then by assumptionPlanetmathPlanetmathf¯(xδ)δ+ϵf¯(xϵ), hencef¯(xδ)(v)δ+ϵf¯(xϵ)(v). Furthermore,f¯(𝗅𝗂𝗆(x)) is non-expanding because f¯(xϵ) is such by inductionhypothesis. Indeed, if uϵv then, for all ϵ:,

f¯(xϵ/3)(u)ϵ/3f¯(xϵ/3)(v),

therefore f¯(𝗅𝗂𝗆(x))(u)ϵf¯(𝗅𝗂𝗆(x))(v) by the fourth constructor of .

We still have to check four more conditions, let us illustrate just one. Supposeϵ:+ and for some δ:+ we have 𝗋𝖺𝗍(q)ϵ-δyδ and f¯(𝗋𝖺𝗍(q))ϵ-δf¯(yδ). To showf¯(𝗋𝖺𝗍(q))ϵf¯(𝗅𝗂𝗆(y)), consider any v:𝖼 and observe that

f¯(𝗋𝖺𝗍(q))(v)ϵ-δf¯(yδ)(v).

Therefore, by the second constructor of , we havef¯(𝗋𝖺𝗍(q))(v)ϵf¯(𝗅𝗂𝗆(y))(v)as required.∎

We may apply \\autorefRC-binary-nonexpanding-extension to any bivariate rational functionMathworldPlanetmathwhich is non-expanding separately in each variable. Addition is such a function, thereforewe get +:𝖼×𝖼𝖼.Furthermore, the extension is unique as long as werequire it to be non-expanding in each variable, and just as in the univariate case,identitiesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on rationals extend to identities on reals. Since composition of non-expandingmaps is again non-expanding, we may conclude that addition satisfies the usual properties,such as commutativity and associativity.Therefore, (𝖼,0,+,-) is a commutativegroup.

We may also apply \\autorefRC-binary-nonexpanding-extension to the functions min:× and max:×, which turns 𝖼 into a latticeMathworldPlanetmathPlanetmath. The partialorderMathworldPlanetmath on 𝖼 is defined in terms of max as

(uv):(max(u,v)=v).

The relationMathworldPlanetmathPlanetmath is a partial order because it is such on , and the axioms of apartial order are expressible as equations in terms of min and max, so they transferto 𝖼.

Another function which extends to 𝖼 by the same method is the absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath ||.Again, it has the expected properties because they transfer from to 𝖼.

From we get the strict order < by

(u<v):(q,r:).(u𝗋𝖺𝗍(q))(q<r)(𝗋𝖺𝗍(r)v).

That is, u<v holds when there merely exists a pair of rational numbers q<r such that x𝗋𝖺𝗍(q) and 𝗋𝖺𝗍(r)v. It is not hard to check that < is irreflexiveMathworldPlanetmath andtransitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and has other properties that are expected for an ordered field.The archimedean principle follows directly from the definition of <.

Theorem 11.3.2 (Archimedean principle for Rc).

For every u,v:Rc such that u<v there merely exists q:Q such that u<q<v.

Proof.

From u<v we merely get r,s: such that ur<sv, and we may take q:(r+s)/2.∎

We now have enough structure on 𝖼 to express uϵv with standard concepts.

Lemma 11.3.3.

If q:Q and u:Rc satisfy urat(q), then for any v:Rc and ϵ:Q+, if uϵv then vrat(q+ϵ).

Proof.

Note that the function max(𝗋𝖺𝗍(q),):𝖼𝖼 is Lipschitz with constant 1.First consider the case when u=𝗋𝖺𝗍(r) is rational.For this we use inductionMathworldPlanetmath on v.If v is rational, then the statement is obvious.If v is 𝗅𝗂𝗆(y), we assume inductively that for any ϵ,δ, if 𝗋𝖺𝗍(r)ϵyδ then yδ𝗋𝖺𝗍(q+ϵ), i.e. max(𝗋𝖺𝗍(q+ϵ),yδ)=𝗋𝖺𝗍(q+ϵ).

Now assuming ϵ and 𝗋𝖺𝗍(r)ϵ𝗅𝗂𝗆(y), we have θ such that 𝗋𝖺𝗍(r)ϵ-θ𝗅𝗂𝗆(y), hence 𝗋𝖺𝗍(r)ϵyδ whenever δ<θ.Thus, the inductive hypothesis gives max(𝗋𝖺𝗍(q+ϵ),yδ)=𝗋𝖺𝗍(q+ϵ) for such δ.But by definition,

max(𝗋𝖺𝗍(q+ϵ),𝗅𝗂𝗆(y))𝗅𝗂𝗆(λδ.max(𝗋𝖺𝗍(q+ϵ),yδ)).

Since the limit of an eventually constant Cauchy approximation is that constant, we have

max(𝗋𝖺𝗍(q+ϵ),𝗅𝗂𝗆(y))=𝗋𝖺𝗍(q+ϵ),

hence 𝗅𝗂𝗆(y)𝗋𝖺𝗍(q+ϵ).

Now consider a general u:𝖼.Since u𝗋𝖺𝗍(q) means max(𝗋𝖺𝗍(q),u)=𝗋𝖺𝗍(q), the assumption uϵv and the Lipschitz property of max(𝗋𝖺𝗍(q),-) imply max(𝗋𝖺𝗍(q),v)ϵ𝗋𝖺𝗍(q).Thus, since 𝗋𝖺𝗍(q)𝗋𝖺𝗍(q), the first case implies max(𝗋𝖺𝗍(q),v)𝗋𝖺𝗍(q+ϵ), and hence v𝗋𝖺𝗍(q+ϵ) by transitivity of .∎

Lemma 11.3.4.

Suppose q:Q and u:Rc satisfy u<rat(q). Then:

  1. 1.

    For any v:𝖼 and ϵ:+, if uϵv then v<𝗋𝖺𝗍(q+ϵ).

  2. 2.

    There exists ϵ:+ such that for any v:𝖼, if uϵv we have v<𝗋𝖺𝗍(q).

Proof.

By definition, u<𝗋𝖺𝗍(q) means there is r: with r<q and u𝗋𝖺𝗍(r).Then by \\autorefthm:RC-le-grow, for any ϵ, if uϵv then v𝗋𝖺𝗍(r+ϵ).ConclusionMathworldPlanetmath 1 follows immediately since r+ϵ<q+ϵ, while for 2 we can take any ϵ<q-r.∎

We are now able to show that the auxiliary relation is what we think it is.

Theorem 11.3.5.

(uϵv)(|u-v|<𝗋𝖺𝗍(ϵ))for all u,v:Rc and ϵ:Q+.

Proof.

The Lipschitz properties of subtractionPlanetmathPlanetmath and absolute value imply that if uϵv, then |u-v|ϵ|u-u|=0.Thus, for the left-to-right direction, it will suffice to show that if uϵ0, then |u|<𝗋𝖺𝗍(ϵ).We proceed by 𝖼-induction on u.

If u is rational, the statement follows immediately since absolute value and order extend the standard ones on +.If u is 𝗅𝗂𝗆(x), then by roundedness we have θ:+ with 𝗅𝗂𝗆(x)ϵ-θ0.By the triangle inequalityMathworldMathworldPlanetmath, therefore, we have xθ/3ϵ-2θ/30, so the inductive hypothesis yields |xθ/3|<𝗋𝖺𝗍(ϵ-2θ/3).But xθ/32θ/3𝗅𝗂𝗆(x), hence |xθ/3|2θ/3|𝗅𝗂𝗆(x)| by the Lipschitz property, so \\autorefthm:RC-lt-open1 implies |𝗅𝗂𝗆(x)|<𝗋𝖺𝗍(ϵ).

In the other direction, we use 𝖼-induction on u and v.If both are rational, this is the first constructor of .

If u is 𝗋𝖺𝗍(q) and v is 𝗅𝗂𝗆(y), we assume inductively that for any ϵ,δ, if |𝗋𝖺𝗍(q)-yδ|<𝗋𝖺𝗍(ϵ) then 𝗋𝖺𝗍(q)ϵyδ.Fix an ϵ such that |𝗋𝖺𝗍(q)-𝗅𝗂𝗆(y)|<𝗋𝖺𝗍(ϵ).Since is order-dense in 𝖼, there exists θ<ϵ with |𝗋𝖺𝗍(q)-𝗅𝗂𝗆(y)|<𝗋𝖺𝗍(θ).Now for any δ,η we have 𝗅𝗂𝗆(y)2δyδ, hence by the Lipschitz property

|𝗋𝖺𝗍(q)-𝗅𝗂𝗆(y)|δ+η|𝗋𝖺𝗍(q)-yδ|.

Thus, by \\autorefthm:RC-lt-open1, we have |𝗋𝖺𝗍(q)-yδ|<𝗋𝖺𝗍(θ+2δ).So by the inductive hypothesis, 𝗋𝖺𝗍(q)θ+2δyδ, and thus 𝗋𝖺𝗍(q)θ+4δ𝗅𝗂𝗆(y) by the triangle inequality.Thus, it suffices to choose δ:(ϵ-θ)/4.

The remaining two cases are entirely analogous.∎

Next, we would like to equip 𝖼 with multiplicative structure. For each q: themap rqr is Lipschitz with constant11We defined Lipschitzconstants as positive rational numbers. |q|+1, and so we can extend it tomultiplication by q on the real numbers. Therefore 𝖼 is a vector space over .In general, we can define multiplication of real numbers as

uv:12((u+v)2-u2-v2),(11.3.6)

so we just need squaring uu2 as a map 𝖼𝖼. Squaring is not aLipschitz map, but it is Lipschitz on every bounded domain, which allows us to patch ittogether. Define the open and closed intervalsMathworldPlanetmath

[u,v]:\\setofx:𝖼|uxv  and  (u,v):\\setofx:𝖼|u<x<v.

Although technically an element of [u,v] or (u,v) is a Cauchy real number together with a proof, since the latter inhabits a mere proposition it is uninteresting.Thus, as is common with subset types, we generally write simply x:[u,v] whenever x:𝖼 is such that uxv, and similarly.

Theorem 11.3.7.

There exists a unique function ()2:RcRc which extends squaring qq2 of rational numbers and satisfies

(n:).(u,v:[-n,n]).|u2-v2|2n|u-v|.
Proof.

We first observe that for every u:𝖼 there merely exists n: such that -nun, see \\autorefex:traditional-archimedean, so the map

e:(n:[-n,n])𝖼  defined by  e(n,x):x

is surjective. Next, for each n:, the squaring map

sn:\\setofq:|-nqn  defined by  sn(q):q2

is Lipschitz with constant 2n, so we can use \\autorefRC-extend-Q-Lipschitz toextend it to a map s¯n:[-n,n]𝖼 with Lipschitz constant 2n, see\\autorefRC-Lipschitz-on-interval for details. The maps s¯n are compatible: ifm<n for some m,n: then sn restricted to [-m,m] must agree with smbecause both are Lipschitz, and therefore continuousMathworldPlanetmathPlanetmath in the senseof \\autorefRC-continuous-eq. Therefore, by \\autoreflem:images_are_coequalizers the map

(n:[-n,n])𝖼,given by  (n,x)sn(x)

factors uniquely through 𝖼 to give us the desired function.∎

At this point we have the ring structure of the reals and the archimedeanPlanetmathPlanetmathPlanetmath order. Toestablish 𝖼 as an archimedean ordered field, we still need inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Theorem 11.3.8.

A Cauchy real is invertible if, and only if, it is apart from zero.

Proof.

First, suppose u:𝖼 has an inverse v:𝖼 By the archimedean principle there is q: such that |v|<q. Then 1=|uv|<|u|v<|u|q and hence |u|>1/q, which is to say that u#0.

For the converseMathworldPlanetmath we construct the inverse map

()-1:\\setofu:𝖼|u#0𝖼

by patching together functions, similarly to the construction of squaring in\\autorefRC-squaring. We only outline the main steps. For every q: let

[q,):\\setofu:𝖼|qu  and  (-,q]:\\setofu:𝖼|u-q.

Then, as q ranges over +, the types (-,q] and [q,) jointly cover\\setofu:𝖼|u#0. On each such [q,) and (-,q] theinverse function is obtained by an application of \\autorefRC-extend-Q-Lipschitzwith Lipschitz constant 1/q2. Finally, \\autoreflem:images_are_coequalizersguarantees that the inverse function factors uniquely through \\setofu:𝖼|u#0.∎

We summarize the algebraic structurePlanetmathPlanetmath of 𝖼 with a theorem.

Theorem 11.3.9.

The Cauchy reals form an archimedean ordered field.

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