Real Analysis – Sequences IV
After having stated and/or proved some important theorems about sequences in the previous post we will know introduce some auxiliary notions that will help us continuing our study of sequences.
— 5.3. Relationships Between Sequences —
Definition 19 Let us consider and . Furthermore let us suppose that there exists another sequence, , such as .
If we’ll say that is asymptotically equal to and denote it by . If we can write . |
As an example let us consider the sequence . It is easy to see that in this case we have .
We can write .
In this case it is and we have .
Let us now try to give a more intuitive meaning to these three notions introduced so far:
First of the notion expresses the fact the difference between and tends to as . That is to say that the two sequences get closer and closer together.
The notion of expresses the fact the both sequences differ only by a scale factor. That is to say that they have the same kind of behavior at .
The meaning of the sentence the same kind of behavior will be made clearer as real analysis gets unfolded in this blog.
The notion of tell us at the gets smaller and smaller when compared to when we get to . In a more formal way: if
Let us now give some examples in order to make things a little bit easier to grasp:
This is easy to see if we write . Taking we see that it is effectively
In this case we write and take . Since is a bounded function we get the intended result.
— 5.4. Final Comments on Sequences —
Definition 21 We’ll say that is a subsequence of whenever is a sequence that tends to . |
Roughly speaking a subsequence, , of a given sequence, , is sequence that doesn’t consider some of the indexes of the initial sequence.
A few examples of subsequences would be (where we don’t take into account the odd numbered indexes of the initial sequence), (only taking into account the the perfect square indexes of the initial sequence).
We already saw that was a converging sequence, then even though appears to be a harder sequence we can say, without any effort, that if we note that it is actually and so is a subsequence of a converging sequence.
Corollary 25 If a sequence has two subsequences with distinct limits then the sequence is divergent. Proof: Follows directly from . |
As an application from the previous corollary we have .
and it is .
and it is .
In conclusion is a divergent sequence.
Theorem 26 (Bolzano-Weierstrass) Each bounded sequence has a converging subsequence in .
Proof: This is only the sketch of a proof. One way to do this is first to prove that all sequences have a monotone subsequence. Applying this result to a bounded sequence we’d have that the bounded sequence have a subsequence that is monotone and bounded (since the sequence is bounded). But by the Corollary 21 we know that a bounded and monotone sequence is convergent. |
Definition 22 Let . We’ll say that is a compact interval if it is bounded and closed. |
Corollary 27 Let be a compact interval and . Then where is a subsequence of .
Proof: Let be the interval and be a sequence of points in . Since , is bounded. From the theorem 26 has a converging subsequence . For it also is . This implies |
April 3, 2009 at 5:11 pm
[…] 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 in the […]
March 30, 2010 at 9:18 am
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Thanks for the comments. It is only a hobby.
March 8, 2014 at 2:51 pm
[…] is a sequence of points in the compact interval (see definition 22 in post Real Analysis – Sequences IV) , by Corollary 27 (also in post Real Analysis – Sequences IV) there […]