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Every infinite decimal sequence is convergent.
Or, in other words, every infinite decimal is a real number. That is to
say, the set of real numbers is the set of limits of infinite decimal
sequences.
31 Write down the first five terms of the infinite decimal sequence with
limit
0.123 456 789 101 112 . . ..
Is the sequence monotonic increasing?
Is every term rational?
State an upper bound for this sequence.
Can you find an upper bound which is smaller than this?
32 If two numbers
a
and
b
are given as infinite decimals, explain how
a
b
,
a
b
,
ab
and
a
/
b
are determined when
b
0.
The remainder of this chapter will be devoted to establishing six
properties of the real numbers, any one of which is in fact equivalent to
the completeness principle, in the context of the number properties
described as (1), (2) and (3) above. Other authors commonly select one
of these six as their aMrmation of completeness, but this list does not
exhaust the alternatives.
I. Every bounded monotonic sequence of real numbers is convergent.
II. The intersection of a set of
nested
closed intervals is not empty (the
term 'nested' is defined in qn 42).
III. Every bounded sequence of real numbers has a convergent
subsequence.
IV. Every infinite bounded set of real numbers has a
cluster point
(the
term is defined before qn 48).
V. Every
Cauchy sequence
of real numbers is convergent (the term
'Cauchy sequence' is defined before qn 56).
VI. Every non-empty set of real numbers which is bounded above has a
least upper bound.
You will find I, III and VI used repeatedly in the rest of the topic,
though all six are necessary for the further study of analysis.
