Homework Help: Two subsequential limit questions

1. Feb 21, 2012

hiro

I have two questions involving subsequential limits. One I have started and understand what I need to do, the second I really don't know how to start.

First question

Let 0 ≤ a < b < +∞. Define the sequence a$_{n}$ recursively by setting a$_{1}$ = a, a$_{2}$ = b, and a$_{n+2}$=(a$_{n}$+a$_{n+1}$)/2 $\forall$n.

Show that the sequences a$_{2n}$ and a$_{2n-1}$ are monotonic and convergent. Does a$_{n}$ converge? To what?

Solution attempt
This is the second part of the question. The first part I already proved, and shows that if a$_{2n}$ and a$_{2n-1}$ both converge to the same limit then a$_{n}$ converges to that limit.

I can show that both subsequences are bounded. I can also show that if their limits exists, they are the same (by plugging in a$_{2n}$→L a$_{2n-1}$→L and using the recurrence relations). The limit is obviously a+(2/3)b (though I need to prove it).

So all I need to do is show that a$_{2n-1}$ is nondecreasing and a$_{2n}$ is nonincreasing. Unfortunately I can't find the right induction proof to show this. Then I need to find one of the limits (either the full sequence or the even or odd subsequence).

Second question
Let λ$\in$[0,1]. Show that there exists a sequence r$_{n}$ such that r$_{n}$→λ. r$_{n}$$\in${0,1/2$^n$,2/2$^n$,...,(2$^n$-1)/2$^n$,1}.

Solution attempt
The correct theorem is presumably:

Let S denote the set of subsequential limits of a sequence s$_{n}$. Suppose t$_{n}$ is a sequence in S$\cap$ℝ and that t$_{n}$→t. Then t belongs to S.

However I can't figure out how to apply it.

2. Feb 21, 2012

alanlu

For the first question, the limit should be a+2(b-a)/3.

Can you write the subsequences in a closed form?

For the second question, let e > 0. What about N such that 1 / 2N < e?

3. Feb 21, 2012

hiro

The numerators look like a Fibonacci type sequence (a Horadam Sequence?). The bottom is just 2^(n-2). You're correct about the limit, I mistyped. Unfortunately I can't figure out if this helps. Is the pattern more obvious if you split off the odd/even sequences? I can't seem to find a pattern in each individual subsequence.

I'm starting to see the proof for the second problem. As I recall if $\exists$N such that $\forall$n≥N |a$_{n+1}$-a$_{n}$|<1/2$^{n}$ then the sequence is Cauchy. Because the sequence would be bounded between 0 and 1 the limit exists and must lie in [0,1]. However, that doesn't prove that any real number in that range can be made the limit of the sequence. Can you help with that bit?

4. Feb 22, 2012

alanlu

The Fibonacci sequence itself can be lifted into a closed form exponential expression, and the subsequences are exponential, or geometric sums. There might be a way to write the whole sequence in a closed form as an alternating geometric sum, now that I think about it. In any case, you want to have a form such that F(1) = a, F(2) = b, F(3) = (a + b) / 2, F(4) = ((a + b) / 2 + b) / 2...

For #2: You want to have for all e > 0 there is an N such that for all n ≥ N, |an+1-an|< 1/2N < e.

Anyway, given real r, how many dyadic rationals of the form i / 2n are you guaranteed to find in the interval (r - 1/2n, r + 1/2n)?

Edit: Also, it may be easier to think about #1 in the interval [0, 1]. Generalizing to [a, b] is fairly trivial.