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

is non-negative, so by the Σ Theorem, the series

∞
x n

10 n
n=1

∞ is a
converges to the least upper bound of the sequence of partial sums. Let’s call this x . If {y n } n=1
different element of S , then by the same reasoning

N
1
y n
< ∀N ∈ N,
10 n 9
n=1
and

y n
≥ 0 ∀n ∈ N,
10 n

so by the Σ Proposition, this series converges to the least upper bound of its sequence of partial sums.
Let’s call this y . By the Decimal Expansion Theorem, the decimal expansions of x and y are unique,
because there are no 9s in these decimal expansions. So for different sequences

∞ ∞
{x n } n=1 = {y n } n=1

the real numbers x = y , because they have different decimal expansions. This means that for each
element of S , there is a unique real number between 0 and 1. So, the set of real numbers contains at
least as many elements as S , and S is uncountable. If R were countable, then it would be possible to

assign a natural number to each of the sequences of S, since they each correspond to one unique real
number. But, because S is uncountable, this is impossible. So R must also be uncountable.

♥

7.5 The Prime Number Theorem

Without the help of analysis, we can only prove that there are infinitely many prime numbers, and that
the set of all prime numbers is countable. But, the set of natural numbers is also infinite and countable,
yet it is much larger than the set of all prime numbers. To better understand how many natural numbers

are prime, we can use a sequence and analysis. Let

P n =the number of primes whichare less than or equalto n.

So,

P 1 =0,

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