Definition.
Let (_{
)
} be a sequence. We say that (_{
)
} is bounded if there are some
This page contains two results on bounds, both very useful. The first says that a convergent sequence is bounded. This is useful because the definition of boundedness is much simpler than the definition of convergence. Convergence has four quantifiers whereas boundedness can be written with only two. It is often easier to show a sequence is not bounded than to show it does not converge to any limit.
The definition of boundedness of sequences is equivalent to
Definition.
Let (_{
)
} be a sequence. Then (_{
)
} is bounded if there is some
Theorem on Boundedness of Convergent Sequences.
Proof.
Remark.
We are given a statement starting
Let
Let
Remark.
We now need to show
Subproof.
Therefore
Therefore
Example.
For many sequences, such as the ones given by
_{
=
} or _{
=(-1)
.
}
it is possible to show the sequence is not bounded and therefore
the sequence is not convergent. That's a saving of two quantifiers'
worth of work. On the other hand, not all non-convergent sequences
are unbounded.
If a sequence is bounded there is the possibility that is has a limit, though this will not always be the case. If it does have a limit, the bound on the sequence also bounds the limit, but there is a catch which you must be careful of.
Theorem giving bounds on limits.
Suppose (_{
)
} is a sequence
that converges to some
The catch is that you cannot conclude that
Proof.
Remark.
We prove each of these inequalities individually by contradiction.
Subproof.
So
Subproof.
So
Example.
Let _{
=1-12
}. So
_{
1
} as
Here is another variation of the same result.
Theorem giving bounds on limits.
Suppose (_{
)
} is a sequence
that converges to some
Proof.
Exercise. Check the proof above works unchanged, in particular
that it only needs