How Does the Invariance Principle Apply to Limits in Engel's Problem?

[ziggyggiz]

Homework Statement

Hi Guys,

This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. I do not know where my knowledge deficits lie, and was recommended this site for help.

"E1. Starting with a point S (a, b) of the plane with 0 < b < a, we generate a sequence of points (xn, yn) according to the rule x0 = a, y0 = b, x_n+1 =(x_n + y_n) / 2 and y_n+1 = (2x_ny_n) / (x_n + y_n).

Here it is easy to find an invariant. From (x_n+1y_n+1) = x_ny_n, for all n we deduce x_ny_n = ab for all n. This is the invariant we are looking for. Initially, we have y0 < x0. This relation also remains invariant. Indeed, suppose yn < xn for some n. Then x_n+1 is the midpoint of the segment with endpoints y_n, x_n. Moreover, y_n+1 < x_n+1 since the harmonic mean is strictly less than the arithmetic mean.

Thus,
0 < x_n+1 − y_n+1 = [(x_n − y_n) / (x_n + y_n)] * [(x_n − y_n) / 2] < (x_n − y_n) / 2
for all n. So we have limx_n = lim y_n = x with x² = ab or x = √ab.

Here the invariant helped us very much, but its recognition was not yet the
solution, although the completion of the solution was trivial."

The Attempt at a Solution

I cannot figure out the bit in bold at all. It says lim x_n = lim y_n, but where does this come from? From a cursory look at the definition of a limt, is it simply since |x_n+1 - y_n+1| < (x_n-y_n)/2, we find that for all N>n, that x_n+1 and y_n+1 are limits of each other?

 

[certainly]

Shouldn't this be in the math section ? I don't see any harmonic motion here....
[EDIT:- I now realize that perhaps you intend those n's and (n+1)'s to be in the subscript, you can easily do that from the post template...]

 

[ziggyggiz]

Shouldn't this be in the math section ? I don't see any harmonic motion here....
[EDIT:- I now realize that perhaps you intend those n's and (n+1)'s to be in the subscript, you can easily do that from the post template...]

I made the changes.

I was not aware this is not harmonic motion question (I assumed the sequence was following a harmonic motion) but I didn't realize there was a precise definition. I will move this to math section.

 

[certainly]

Quite alright. "To err is to human" eh!
Cheers :)

 

[ziggyggiz]

Quite alright. "To err is to human" eh!
Cheers :)

Thanks

 

[certainly]

Now, do you know anything about limits?

 

[certainly]

0 < xn+1 − yn+1 = [(xn − yn) / (xn + yn)] * [(xn − yn) / 2] < (xn − yn) / 2

Think geometrically. What is feature of the two points $x_{n+1}$ and $y_{n+1}$ is different (or has changed) from the points $x_n$ and $y_n$
[EDIT:- and what happens to this "feature" if you keep creating these new points ad infinitum ? that is to say what happens "in the limit".]

 

[ziggyggiz]

X_n gets larger while Y_n gets smaller to preserve the fact that X_nY_n =ab right?

 

[certainly]

Precisely...now if you keep doing this what will happen eventually i.e "what will happen in the limit" ?
[EDIT:- Geometrically speaking you can imagine $X_n$ and $Y_n$ as two points slowly approaching each other from opposite directions on the number line, one gets bigger, the other gets smaller... they keep getting closer to one another after each iteration]

 

[ziggyggiz]

Now, do you know anything about limits?

Not much I am afraid. I know that a sequence goes to a limit if it gets closer and closer as the index goes up

 

[certainly]

It will get closer and closer to something for the limit to exist, otherwise it would just go to infinity... am I right?
[EDIT:- now think of what that "something" is in the original question. Also see edit to post #9]

 

[ziggyggiz]

Precisely...now if you keep doing this what will happen eventually i.e "what will happen in the limit" ?
[EDIT:- Geometrically speaking you can imagine $X_n$ and $Y_n$ as two points slowly approaching each other from opposite directions on the number line, one gets bigger, the other gets smaller... they keep getting closer to one another after each iteration]

Ah ok! I get it now, they will approach each other due to the invariant nature of their relation ship X_nY_n=ab. So their limit can be denoted arbitrarily by X. And the limit of their products is then just the product of their limits which implies X² = ab

Thank you for your help you have helped in my journey; much appreciated!

 

[certainly]

Well done...you solved it.
Also this was a good intro to limits ;)