You are all familiar with decimals. In the United States these are
written in the form .a1a2a3
where the ai
are integers between 0 and
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
Now what is meant by a nonterminating decimal?
Definition 4.9.10 Let .a1a2
be a decimal. Define
Proposition 4.9.11 The above definition makes sense. Also every number in
can be written as such a decimal.
Proof: Note the sequence
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.
Therefore, since this holds for all n, it follows the above sequence is bounded above. It follows
the limit exists.
Now suppose x ∈ [0,1). Let
≤ x <
is an integer between 0 and 9. If
each between 0 and 9 have been obtained such that
k=10 ≡ 0). Then from the above,
and so there exists an+1 such that
which shows that
because the distance between the partial sum up to n and x is always no more than 1∕10n. In
case x = 1, just let each an = 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 4.9.12 The 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,
Consider the diagonal decimal,
Now define a decimal expansion for another number in [0,1) as follows.
Thus y is not equal to any of the xk which is a contradiction since y ∈ [0,1). ■