# Proving well definedness of addition in real numbers. Real numbers defined as infinite decimal expansions.

As the title says. I have to prove that the addition in the real numbers is well defined. Here are the definitions of both the real numbers and the addition of real numbers. These are translations.

Definition of real numbers

We define non-negative real numbers as infinite decimal expressions of the form

$$A.a_{1}a_{2}a_{3...,}$$

where $A\in\mathbb{N}\cup\{0\}$ and $a_{j}\in\left\{ 0,1,...,9\right\}$. The ellipsis indicate that there is an infinite number of $a_{j}$ .We also suppose that $\forall n\in\mathbb{N},\exists m>n$ such that $a_{m}\neq9$

Let D be a subset of $\mathbb{Q}$ , composed of numbers of the form

$$\frac{a}{10^{n}},a\in\mathbb{Z}$$

Let $\alpha,\beta\in\mathbb{R},$

$$U=\left\{ x\in D\mid x\leqslant\alpha\right\}$$ ,

$$V=\left\{ y\in D\mid y\leqslant\beta\right\}$$ ,

$$C=\left\{ x+y\mid x\in U,y\in V\right\}$$

we define $\alpha+\beta$ as the supremum of C.

Definition of supremum

Let $S \subset \mathbb{R}$ and $\alpha \in \mathbb{R}$. $\alpha$ is the supremum of S if

i) $\alpha$ is an upper bound of S

ii) if $\beta$ is an upper bound of S, then $\alpha \le \beta$

It has already been proven that the supremum is unique

Question: Prove that addition is well defined.

I don't know whether the question makes sense. I thought proving that something is “well defined” meant proving that using different representations of the same number doesn't affect the outcome of the operation. But I only see one representation of a real number here. Does the question make sense? My idea was to prove that sup C has a decimal expansion. Would this be the right thing to prove?

• How is the "supremum" defined in this setup? – Christian Blatter May 23 '16 at 8:27
• @Christian Blatter. I added the definition of supremum. – n4z May 23 '16 at 16:43
• " I thought proving that something is “well defined” meant proving that using different representations of the same number doesn't affect the outcome of the operation." Not entirely. You have to prove 1) that the supremum of C exists. 2) That it is unique. And I suppose if your definition of Real numbers is infinite decimal expansions, that the suprememum is a real number. Have you proven that in the real numbers the suprememum of all bounded sets exist and is real? – fleablood May 23 '16 at 18:20

My impression is that your system ${\mathbb R}$ of real numbers is a set of abstract entities satisfying the usual axioms. You are told to prove that the set of decimal fractions $A.a_1a_2a_3\ldots\$, provided with algorithms for addition and multiplication of such fractions, can serve as a model of ${\mathbb R}$. This is a long story. It is true that $$\sup C=:\gamma\in{\mathbb R}\ ,$$ where $C$ is the set of all sums of finite decimal fractions $a+b$ approximating $\alpha+\beta$, is the abstract object that represents $\alpha+\beta$. But a lot of work is required in order to show that this real number $\gamma$ has an essentially unique decimal representation, which is in some algorithmic relation to the representations of $\alpha$ and $\beta$.