Decimal representation of reals

Definition: Decimal expression

A decimal expression $m.a_1{a}_2{a}_3...$ is a natural $m\in \mathbb{N}$ and a sequence of digits $a_i\in \{0,1,\cdots ,9\}$, which can be written as the sum

$$m+\sum _{i=1}^{\infty }\frac{a_i}{10^i}.$$

• The real number associated with a decimal representation is formally the least upper bound of the set of all finite decimal approximations

$$m.a_1{a}_2{a}_3...=\text{lub}\{m+\sum _{i=1}^n\frac{a_i}{10^i}\mid i\in \mathbb{N}\}.$$

• (Proposition 1.4) Every real number $x\ge 0$ has a decimal representation.
  • (Lemma 1.5) We have $\text{glb}\{10^{-i}\mid i\in \mathbb{N}\}=0$.
  • (Corollary 1.6) We have $\mathrm{0.999...}=1$.
  • (Lemma 1.6) If $x\in \mathbb{R}$ has $x\ge 0$, the set of natural numbers ${\mathbb{N}}_{\le x}=\{n\in \mathbb{N}\mid n\le x\}$ is finite.

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Review §

• Explain why a set that has a least upper bound must be non-empty. ⭐
• Prove Corollary 1.6. ⭐
• Prove Lemma 1.6.
• Prove the decimal representation is unique. ⭐
• Prove that every real number $x\ge 0$ has a decimal representation. ✨
  • Hint: The two sticking points of this proof are showing that the reals are not infinitely small or infinitely large, which are resolved in Lemma 1.5 and 1.6 respectively.