Date: Feb 2, 2013 11:33 PM
Author: mina_world
Subject: Re: Quadruple with (a, b, c, d)
"mina_world" wrote:

>

>Hello teacher~

>

>Suppose not all four integers a, b, c, d are equal.

>Start with (a, b, c, d) and repeatedly replace (a, b, c, d)

>by (a-b, b-c, c-d, d-a).

>

>Then at least one number of the quadruple will eventually

>become arbitrarily large.

>

>-----------------------------------------------------------

>Solution:

>

>Let P_n = (a_n, b_n, c_n, d_n) be the quadruple after

>n iterations.

>

>Then a_n + b_n + c_n + d_n = 0 for n >= 1.

>

>A very importand function for the point P_n in 4-space

>is the square of its distance from the origin (0,0,0,0),

>which is (a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2.

>

>....

>omission

>For reference, text copy with jpg.

>http://board-2.blueweb.co.kr/user/math565/data/math/olilim.jpg

>...

>

>for n >=2,

>(a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2

>>= {2^(n-1)}*{(a_1)^2 + (b_1)^2 + (c_1)^2 + (d_1)^2}.

>

>The distance of the points P_n from the origin increases

>without bound, which means that at least one component must

>become arbitrarily large

>

>----------------------------------------------------------------

>Hm, my question is...

>

>I know that a_n + b_n + c_n + d_n = 0

>and

>I know that (a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2 increases

>without bound.

>

>I can't understand that "at least one component must become

>arbitrarily large". I need your logical explanation or proof.

Let

S_a = {a_n | n in N}

S_b = {b_n | n in N}

S_c = {c_n | n in N}

S_d = {d_n | n in N}

The goal is to show that at least one of the sets

S_a, S_b, S_c, S_d

is unbounded above.

Since the set

{(a_n)^2 + (b_n)^2 + (c_n)^2 + (d_n)^2 | n in N}

is unbounded, it follows that at least one of the sets

S_a, S_b, S_c, S_d

is unbounded.

Without loss of generality, assume S_a is unbounded.

If S_a is unbounded above, we're done, so assume instead

that S_a is bounded above.

Since S_a is unbounded, S_a must be unbounded below.

Then

a_n + b_n + c_n + d_n = 0 for all n in N

=> b_n + c_n + d_n = -(a_n) for all n in N

=> {b_n + c_n + d_n | n in N} is unbounded above

=> at least one of S_b, S_c, S_d is unbounded above

so at least one of S_a, S_b, S_c, S_d is unbounded above,

as was to be shown.

Oh, thank you very much for your detailed explanation.