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I'm writing a proof that the real numbers are uncountable. The proof proceeds by contradiction, assuming $f: \mathbb{N} \to \mathbb{R}$ is a bijection. Then $f(n) = a_0. a_1 a_2 a_3 \ldots$. For each position after the decimal point, it defines the set $S_n$ to be the number of elements that exist in all possible decimal expansions of $f(n)$. The claim is that $|S_n| \leq 2$.

I cannot figure out how to prove this rigorously, and I feel my knowledge of this problem is very superficial. Some real numbers, like $\pi$, don't terminate at all or repeat, so they have exactly one decimal expansion. A number like $0.99\overline{9}$ have two decimal expansions, namely $0.99\overline{9} = 1$. I can also take a decimal that terminates like $0.234$ and rewrite it as $0.23\overline{9}$. (I'm not sure if an infinite string of $1$'s corresponds to a $2$, infinite string of $3$'s corresponds to a $4$, and so forth, or this is particular to $9$'s).

How can I justify this fact?

Martin Sleziak
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JohnT
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1 Answers1

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In order to understand why any real number has at most two decimal expansions, it is important to keep in mind what a decimal expansion is, i.e. the limit of the power series (base $10$) of the string of digits in it. For now, it suffices to only consider the real numbers in $[0,1)$, as any other real number is simply an integer added to an element of this set.

Let's start with two decimal expansions $0.a_1 a_2 a_3 ...$ and $0.b_1 b_2 b_3 ...$. We wish to examine when they correspond to the same real number. Let $k$ be the smallest positive integer such that $a_k\neq b_k$. Such a $k$ must exist unless $\: a_i=b_i\;\forall \; i\in \mathbb{N}$. Without loss of generality, say $a_k>b_k$, then in fact $\;a_k\geq b_k + 1$. Keep in mind that $\: a_n=b_n\;\forall \; n<k$. Keeping $b_1, b_2... b_k$ fixed,

  • What is the largest value $0.b_1 b_2 b_3 ...$ can attain?
  • Can this value ever be greater than $0.a_1 a_2 a_3 ...$?
  • When can it be equal to $0.a_1 a_2 a_3 ...$?

You may answer these questions by simply knowing the sum of an infinite geometric series. It should then be easy to see why every non-terminating decimal expansion (like $\pi$ as you pointed out, or even $\frac{1}{3}$) is unique, and why every real with a terminating decimal expansion (like $0.234000...$) has exactly two decimal representations. Feel free to ask follow up questions if you are stuck anywhere!

SlimWiz
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