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Blast into Math! Analatic nummer theora: ants, ghosts and giants

So, by definition the sequence of partial sums

N
x n

10 n
n=1

converges to x .

♥
Using decimal expansions we can prove that the set of real numbers is uncountable.

7.4.1 Uncountably many real numbers

The set of real numbers contains Q as a subset, so R has infinitely many elements. All the sets N, Z
and Q are countable, but the set of real numbers is uncountable. This means that there are too many

real numbers to associate a natural number to each real number. To prove this fact, we will first prove
that a certain set is uncountable.

Proposition 7.4.2 (Uncountable Set Proposition). Let S be the set of all sequences whose elements are
each either 0 or 1. Then S has infinitely many elements and is uncountable.

Proof: First, let’s prove that S has infinitely many elements. Since S contains all sequences whose
elements are 0 and 1, we can prove that S has infinitely many elements by finding infinitely many
sequences of 0 and 1. The sequence

{1, 0, 0, 0, ....}

whose first element is 1 and all other elements are 0 is contained in S. So is the sequence

{0, 1, 0, 0, ...},

whose second element is 1 and all other elements are 0 contained in S. For each n ∈ N, the sequence
whose n element is 1 and all other elements are 0 is contained in S. Since each of these is a different
th
sequence, this means that S contains at least as many elements as N, and is therefore infinite.

The definition of uncountable is not countable, so it makes sense to prove the proposition by contradiction.
Let’s assume that S is countable. This would mean that we can associate to each sequence in S a natural
number. The first sequence we shall call s 1 , and the second sequence s 2 , and so forth. Now, we will
construct a sequence

∞
{x n } n=1 ,

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