Proposition: Addition of Real Numbers

Any real number \(x\in\mathbb R\) is by the corresponding proposition an equivalence class \(x:=(x_n)_{n\in\mathbb N} + I,\) where \((x_n)_{n\in\mathbb N}\) denotes a rational Cauchy sequence representing the real number \(x\), and where \(I\) denotes the set of all rational sequences, which converge to \(0\).

Given two real numbers \(x=(x_n)_{n\in\mathbb N}+ I\) and \(y=(y_n)_{n\in\mathbb N}+ I\), the addition of real numbers is defined by

\[\begin{array}{rcccl} x+y&:=&(x_n+y_n)_{n\in\mathbb N}+ I, \end{array}\]

where the result is the real number \((x_n + y_n)_{n\in\mathbb N}+ I \), called the sum of the real numbers \(x\) and \(y\). The sum exists and is well-defined, i.e. it does not depend on the specific representatives \((x_n)_{n\in\mathbb N}\) and \((y_n)_{n\in\mathbb N}\) of \(x\) and \(y\).

Proofs: 1

  1. Proposition: Addition Of Real Numbers Is Associative
  2. Proposition: Addition Of Real Numbers Is Commutative
  3. Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers)
  4. Proposition: Existence of Inverse Real Numbers With Respect to Addition
  5. Proposition: Uniqueness of Real Zero
  6. Proposition: Uniqueness of Negative Numbers
  7. Proposition: Addition of Real Numbers Is Cancellative

Definitions: 1 2 3
Parts: 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15 16
Propositions: 17 18 19 20 21 22 23 24 25 26

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013