This web page gives a very basic (almost obvious) theorem that seems at first to be uninteresting, but is in fact one of the keystones that the theory we will develop relies on. Also, together with later material on subsequences, it is one of the key methods for proving a sequence doesn't converge.
As with all the other key definitions and results you should at a minimum learn the statement of this theorem, and ideally learn the proof too.
The theorem on the uniqueness of limits says that
a sequence (_{
)
} can have at most one limit.
In other words, if _{
} and _{
}
then
Theorem on Uniqueness of Limits.
As a consequence, to show that a sequence (_{
)
}
does not converge to some number obvious
is a dangerous
word as occasionally obvious
things turn out to be false).
We will improve upon this idea considerably later on when we
discuss subsequences so I don't provide
any examples here at this stage.
Proof.
Remark.
As before, comments written in this font at a smaller size are not part
of the proof but comments indicating some feature of the proof or how I was thinking about
it. In this proof we'll assume that (_{
)
} converges to both
Remark.
To make use of these last two statements we need to give them positive values
of
Subproof.
Let
Remark.
Note that there is no reason to expect that
Let
But this means that
This completes the proof.