Real Analysis – Sequences III
Theorem 18 Theorem 17 Let be a set of real numbers and . Then there exists a sequence, ,with range in such as .
One can also formulate an analogous theorem for . Proof: Omitted. |
This theorem tell us the algebraic properties of limits of sequences and none of them seems to be too surprising to justify a presentation of all of the proofs.
Property 2 was shown in order for us to gain some more experience with the notions.
In case you are wondering why we used in the limit conditions of both and instead of you have to realise that what matters in the definition of limit is that the distance between the terms of the sequence and its limit has to be smaller and smaller.
If we denote this distance by , or , or , or even is just a matter of convenience.
Proper care is needed with the reading of the enunciate of this theorem. This is just a sufficient condition for a given sequence to have a limit. In no way the converse of the previous theorem (every sequence that has a limit in is a monotone sequence) is a true statement. One has only to think about which tends to and we see that it is not a monotone sequence even though it has a limit.
Corollary 21 Every bounded and monotone sequence is convergent in .
Proof: By hypothesis we know that . By the previous theorem we know that has a limit in . And by the Corollary 15 it is . Hence where . |
Now this corollary here is a slobber-knocker for practical applications. Of course that, if the given sequence converges, we can’t say a thing about to what value it converges to.
But just think about the the fact the we are now able to determine the nature of a sequence without calculating any limits at all. It is now a matter of what is easier to do and/or what we need to know.
In some cases we may need the value of the limit but in other cases just knowing the behavior of the sequence is enough. And of course it also matters if it is more practical to actually calculate the limit and see if it exists and is a real number or not.
For instance given what should be our strategy? Go for the limit or try to prove that we have a bounded and monotone sequence?
Let us do some graphical inspection:
From the graph we can see that appears to bounded by and is increasing, thus monotone.
I use the expression appears because a graphical representation can only contain a finite number of terms and one can’t be sure that something strange doesn’t happen at the points we aren’t representing.
But I’ll try to prove those two propositions anyway and conclude that is indeed a convergent sequence.
Proposition 22 is a convergent sequence.
Proof: First we’ll prove that is increasing. In order to that we’ll calculate .
For us to proceed here we have to remember Bernoulli’s inequality and . This inequality can be proven (and will be in a future date) using mathematical induction. Continuing.
In conclusion and is monotone. Now all we have to do is to prove that is bounded and we’ll know that is convergent. . From what we have already proven we also know that is increasing, so . Now we have to prove that also has an upper bound for it to be bounded. As was proven in here we can write . Writing out the terms we have:
We know that and that . Using those two inequalities (in that respective order) we have:
So what we have is: . Since it is
Tidying up what we have is . Hence is monotone and bounded. Which amounts to being convergent. Furthermore from we know that . This isn’t really much of a help to what value the sequence tends to but at least it is some information. |
February 16, 2009 at 4:20 pm
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April 3, 2009 at 5:11 pm
[…] 3, 2009 After introducing sequences and gaining some knowledge of some of their properties (I,II ,III , and IV) we are ready to embark on the study of real analysis while using concepts that are more […]