# Sequence of real numbers | Proof of convergence

1. Jan 23, 2010

### kingwinner

1. The problem statement, all variables and given/known data

2. Relevant equations
N/A

3. The attempt at a solution
Assuming the truth of part a, I proved part b.
But now I have no idea how to prove parts a & c.
Part a seems true intuitively. The sqaure root of a number between 0 and 1 is will be larger than that number, and if we take more and more square roots, it will get close to 1, and then if we add two numbers that are close to 1, it must be ≥1.
But how can we write a FORMAL proof of it? How can we find/construct N and demonstrate exactly that there exists an N such that n≥N => an≥1?

Any help is much appreciated!

[note: also under discussion in Math Links forum]

Last edited: Jan 23, 2010
2. Jan 25, 2010

### kingwinner

For part a, I have some more idea...if a(n) and a(n+1) are positive then we can surely pick an m such that 1/(2m) is less than them both: just select a sufficiently large m, e.g. select m such that 2m ≥ 1/min{a(n),a(n+1)}.

But how can I find N such that n≥N implies a(n)≥1 ?

Any help is much appreciated!!!

3. Jan 25, 2010

### sutupidmath

Since you seem to have spent quite some time on the problem i will try to give you some hints, i hope i don't get another warning from pf moderators for offering too much help(solving >90% of the problem for the op) :(

This might not be the nicest proof in the world, but i think it works.

As you have figured out the main problem is when 0<a_1<1 and 0<a_o<1. So we will deal with this case only, since others are trivial.
Let:
$$0<a_0<1,0<a_1<1$$
then:
$$a_o<\sqrt{a_o}...and...a_1<\sqrt{a_1}$$

$$a_3=\sqrt{a_0}+\sqrt{a_1}>a_0+a_1$$

If we continue in this fashion, after n-2 steps we would get something like:

$$a_{n+2}=\sqrt{a_{n+1}}+\sqrt{a_n}>a_0+a_1+...+a_n>n*min\{a_0,a_1,...,a_n\}=n*a$$

So, now you see that if we let n>N=1/a we get our result. where a=min{a_o,...,a_n}

cheers!

Last edited: Jan 25, 2010
4. Jan 25, 2010

### kingwinner

Thanks.
Using part a, I proved part b.

5. Jan 26, 2010

### kingwinner

For part c, I'm stuck with using the hint.

From part b, en+2 ≤ (en+1 + en)/3 for n≥N.

In part c, I think I need to end up proving something like
en ≤ (2/3)some exponent involving n max(eN,eN+1)
If the RHS tends to 0, then by squeeze theorem en->0.

But I'm not sure how to SET UP the iteration. From part b, en+2 ≤ (en+1 + en)/3 for n≥N. Is it also true that en+1 ≤ (en + en-1)/3? Why or why not?
And how can I find that "some exponent involving n"?