Proving Convergence of a Subsequence in a Bounded Sequence

[rudders93]

Homework Statement

Without using logarithms, prove that $a^{n}\rightarrow0$ as $n\rightarrow\infty$ for $|a|<1$ by using the properties of the sequence $u_{n}=|a|^{n}$ and its subsequence $u_{2n}$.

The Attempt at a Solution

I'm really stuck on this. One thing I thought of doing was to prove that the sequence is strictly decreasing as $u_{n+1}-u_{n}=|a|^{n+1}-|a|^{n}=|a|^{n}(|a|^{n}-1)<0$ as $|a|<1$. Also we know it's bounded within interval (0,1) and so by the Monotonic Sequence theorem it must converge to some limit. Then using standard limits we can prove that it converges to 0, but I think that's kind of cheating the proof

However, not sure how to do this using the properties of subsequences. How can I do it using that?

Thanks!

 

[HallsofIvy]

Yes, that is a decreasing sequence bounded below by 0 and so, as you say, converges. The point of "subsequences" is this- a sequemce converges, to limit L, if and only if every subseqence converges to L. You know this sequence converges so if any subsequence converges to 0, the entire sequence does. Can you show that $a_{2n}$ converges to 0? What is special about that sequence? In particular, why "2n"?

 

[Dickfore]

You should use this:

http://en.wikipedia.org/wiki/Bernoulli's_inequality

and:

http://en.wikipedia.org/wiki/Squeeze_theorem

in your proof.

Hint: If $|a| < 1$, we can represent it as:
$$|a| = \frac{1}{1 + h}$$
What can you say about h?

 

[rudders93]

Hi, thanks for reply.

Yeah I tried to see find the reason behind the 2n but the only thing I could think of was that it converges at a faster rate than just n, as $u_{2n}=(|a|^{n})^{2}$ but I still couldn't see what I could do to show it converges to 0, other than the standard limits :(

An $\epsilon-N$ could I guess show it converges to 0, but then again, it could do that for just $u_{n}$?

Thanks again for reply

 

[Dickfore]

I think the point is that $u_{2 n} = (u_{n})^{2}$. Since the sequence is convergent, so is any of its subsequences and it has the same limit. Take the limit in the above equality.

 

[rudders93]

Ah i see that, thanks Dickfore! So can do:

$|a|=\frac{1}{1+h}$ so therefore if we take $u_{2n}=|a|^{2}=(\frac{1}{1+h})^{2}=\frac{1}{(1+h^{2})}$ which tends to 0 as n tends to infinity by standard limits. Is that it? But nevertheless, won't just $u_{n}$ also tend to infinity using the same argument, although it may take longer to do so?

Thanks again!

 

[Dickfore]

No, that is not what I meant. My last post was not connected with my previous hint, since you are supposed to follow a different method.

Also, you have a mistake:
$$\left(\frac{1}{1 + h}\right)^{2} \neq \frac{1}{1 + h^{2}}$$

 

[rudders93]

Sorry, I'm not sure how $u_{2n}=(un)^{2}$ can help me :( Nor how the squeeze theorem/bernoulli's can help with in proving that $u_{2n}$ converges without using standard limits :(

 

[Dickfore]

Disregard post #3.

1) Did you prove that $\lbrace u_{n} \rbrace$ is convergent?

2) What does that tell you about the convergence of the subsequence $\lbrace u_{2 n}\rbrace$

3) Can you prove this $u_{2 n} = (u_{n})^{2}$ for the particular sequence we are discussing?

4) What will happen if we take the limit in the equality:

$$\lim_{n \rightarrow \infty}{u_{2 n}} = \lim_{n \rightarrow \infty}{(u_{n})^{2}}$$

 

[rudders93]

ooops sorry that was me being incompetent in latex. $u_{2n}=|a|^{2}=(\frac{1}{1+h})^{2}=\frac{1}{(1+h)^{2}}$. But that nevertheless converges to infinity also by standard limits?

 

[Dickfore]

ooops sorry that was me being incompetent in latex. $u_{2n}=|a|^{2}=(\frac{1}{1+h})^{2}=\frac{1}{(1+h)^{2}}$. But that nevertheless converges to infinity also by standard limits?

What you wrote is a constant sequence, so to speak of its convergence properties is trivial. However, this has nothing to do with the problem we are discussing.

 

[rudders93]

Hmm. Well:

1) Yes ${u_{n}}$ is convergent as it is decreasing and it is bounded as it must converge.

2) Subsequences of convergent sequences must converge, so yes this also converges.

3) Yes. $u_{2n}=|a|^{2n}=(|a|^{n})^{2}=(u_{n})^{2}$

4) Not sure, they're equivalent? If so, how does that help with proving it converges using the subsequence

Thanks again for your patience!

 

[Dickfore]

If the limit of a sequence is L, what is the limit of any of its subsequences?

 

[rudders93]

It will also be L. But I can see how to prove that it converges to L=0 by the limit theorems with $u_{2n}$ but I think you could still prove that same thing with just $u_{n}$?

 

[Dickfore]

But I can see how to prove that it converges to L=0 by the limit theorems with $u_{2n}$ ...

How?

 

[rudders93]

Well:

$u_{2n}=|a|^{2}=(\frac{1}{1+h})^{2}=\frac{1}{(1+h)^{2}}$

So: $$\lim_{n \rightarrow \infty}{(u_{n})^{2}}=\lim_{n \rightarrow \infty}\frac{1}{(1+h)^{2}}=0$$ by standard limits? Or is that incorrect, as that would work just fine using the same argument just for $u_{n}$?

 

[Dickfore]

No, this is wrong.

 

[rudders93]

Hmm, I see that yep my post was incorrect with the limits. I do not know then :(

 

[Dickfore]

All the hints have been given to you. There is no spoon-feeding alowed on this forum. I cannot provide any further insight. I suggest you re-read the thread.

 

[rudders93]

Ok fair enough.

Well, I figured that $0<|a|^{2n}<|a|^{n}$ by using $|a|=\frac{1}{1+h}$ and then applying bernoullis, which results in that as h must be positive to satisfy the upper bound.

From taht I guess you can use standard limits as $|a|^{n}$ converges to 0 by standard limits and so too must [itex]|a|^{2n}[\itex] by the squeeze theorem. But that's once again using the standard limits, which I'm not supposed to use?

 

[Dickfore]

You have not proven what is required in your problem by any of the methods suggested.

 

[AKG]

First of all, forget all about $\frac{1}{1+h}$. Summarize for us what you know so far about the sequence $u_n = a^n$ and the subsequence $u_{2n} = a^{2n}$.

 

[rudders93]

Ok, took some time and I had to look at some other examples. Saw something where you let the limit be L and then you solve algebraically, thought I could apply that, so here it goes:

I know that $u_{2n}$ is a subsequence of $u_{n}$ where $u_{2n}=(a^{n})^{2}$

I'm not to sure about, but is it Bolzano-Weierstrass Theorem that states that the subsequence converges to the same limit as the sequence (I know it says that each bounded real sequence contains a convergent subsequence, but does it also imply that this subsequence converges to the same limit, or is that another theorem? Couldn't find it, just vaguely recall there being one).

Assuming the above is true, then we can assume that given some $L\in\Re$ we have $$\lim_{n \rightarrow \infty}{u_{2n}}=\lim_{n \rightarrow \infty}{u_{n}}=L$$
Hence as $$\lim_{n \rightarrow \infty}{u_{2n}}=\lim_{n \rightarrow \infty}{(u_{n})^{2}}$$
it follows that $$\lim_{n \rightarrow \infty}{(u_{n})^{2}}=\lim_{n \rightarrow \infty}{u_{n}}$$
and so if we sub in L from above we have: $L-L^{2}=0 \Rightarrow L(1-L)=0$ and hence it must be that the limit is either 0 or 1.

From here I'm much more iffy. I think possible that here we can use that $u_{n}$ is a monotonically decreasing sequence as proven in my first post and so given that $|a|<1$ it must be that $u_{n}$ converges to 0 as required?

Is that kind of the proof required?

 

[Dickfore]

Yes, congratulations.

 

[rudders93]

Awesome, thanks!

Just to confirm, what's the name of the theorem that states that a subseqence converges to the same limit as the sequence?

 

[Dickfore]

It does not have a name. It is almost trivial by the definition of a subsequence and a convergent sequence.

 

[rudders93]

Ok awesome, thanks again!