We have already seen the definition
of montonic sequences and the fact that in any Archimedean ordered field,
every number has a monotonic nondecreasing sequence
of rationals converging to it. Monotonic sequences are particularly
straightforward to work with and are the key to stating and understanding
the completeness axiom for the reals. This page sets out some simple
properties of monotonic sequences that will be useful later on.
We shall continue to work with the set of real numbers
The theorem of this section is paraphrased by the title above. There are two versions of the result, one for nonincreasing sequences and one for nondecreasing sequences.
Theorem.
Proof.
A similar result for nonincreasing sequences is proved in exactly the same way.
Theorem.
Proof.
Exercise.
We now look at when two monotonic sequences converge to the same limit. The next theorem provides information on this.
Theorem.
Proof.
Let
Let
Then
This is impossible as, by assumption, there is _{
}
with _{
}. This means that
There is a similar result for nonincreasing sequences.
Theorem.
Proof.
Exercise.
If (_{ ) } is a monotonic nondecreasing sequence and is convergent, then this sequence is bounded, since all convergent sequences are bounded. The converse of this statement is in fact true for the reals, but cannot be proved from the axioms of Archimedean ordered fields. This converse is called the Monotone Convergence Theorem and is discussed in a later web-page.
The point of this short section is that by a theorem above, a monotonic
nondecreasing sequence is bounded above by its limit
Theorem.
The point is that the condition
The results on this web page are not particularly difficult. They are even results that you might have guessed anyway. It does save a small about of time to remember them and use them when you need them, rather than repeat the proofs or the ideas in the proofs, but the choice is yours.