You are all familiar with decimals. In the United States these are
written in the form .a_{1}a_{2}a_{3}

⋅⋅⋅

where the a_{i} are integers between 0 and
9.^{2}
Thus .23417432 is a number written as a decimal. You also recall the meaning of such
notation in the case of a terminating decimal. For example, .234 is defined as

2
10

+

3
102

+

4
103

.
Now what is meant by a nonterminating decimal?

Definition 4.9.10Let .a_{1}a_{2}

⋅⋅⋅

be a decimal. Define

∑n ak-
.a1a2⋅⋅⋅ ≡ nli→m∞ 10k.
k=1

Proposition 4.9.11The above definition makes sense. Also every number in

[0,1]

can be writtenas such a decimal.

Proof: Note the sequence

{∑n -ak-}
k=110k

_{n=1}^{∞} is an increasing sequence. Therefore, if there
exists an upper bound, it follows from Theorem 4.9.7 that this sequence converges and so the
definition is well defined.

because the distance between the partial sum up to n and x is always no more than 1∕10^{n}. In
case x = 1, just let each a_{n} = 9 and observe that the sum of the geometric series equals 1.
■

An amusing application of the above is in the following theorem. It gives an easy way to
verify that the unit interval is uncountable.

Theorem 4.9.12The interval [0,1) is not countable.

Proof: Suppose it were. Then there would exist a list of all the numbers in this interval.
Writing these as decimals,

x ≡ .a a a a a ⋅⋅⋅
x1≡ .a11a12a13a14a15⋅⋅⋅
x2≡ .a21a22a23a14a25⋅⋅⋅
3 31 32.3334 35
..

Consider the diagonal decimal,

.a11a22a33a44⋅⋅⋅

Now define a decimal expansion for another number in [0,1) as follows.

y ≡ .b1b2b3b4⋅⋅⋅

where

|bk − akk|

≥ 4. Then

|y − xk| ≥-4-
10k

Thus y is not equal to any of the x_{k} which is a contradiction since y ∈ [0,1). ■