Two subsequential limit questions

[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.

Thanks for your help.

 

[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 / 2^N < e?

 

[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?

 

[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, |a_n+1-a_n|< 1/2^N < 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.