```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_dis 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_dis unbounded. Without loss of generality, assume S_a is unbounded.If S_a is unbounded above, we're done, so assume insteadthat 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 aboveso 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.
```