Re: Analysis
- From: Kira Yamato <kirakun@xxxxxxxxxxxxx>
- Date: Mon, 22 Oct 2007 17:07:06 -0400
On 2007-10-22 16:50:44 -0400, quasi <quasi@xxxxxxxx> said:
On Mon, 22 Oct 2007 04:56:48 -0400, Kira Yamato
<kirakun@xxxxxxxxxxxxx> wrote:
On 2007-10-20 16:47:49 -0400, quasi <quasi@xxxxxxxx> said:
On 20 Oct 2007 11:14:34 -0700, Kenshin <rurouni_sohjiro@xxxxxxxxxxx>
wrote:
Prove or disprove that there exists A = {a_n l n= 1,2,,,} = Q
(rational number set)
such that sum{ (a_(n+1) - a_n)^2 } converges.
Let x_1, x_2, x_3, ... be any enumeration of Q.
Build a_1, a_2, a_3, ... in such a way as to satisfy the following
conditions:
(1) a_1 = x_1
(2) for all n,
| a_(n+1) - a_n | <= 1/n
(3) Once a_1, a_2, a_3, ... hits x_n, the sequence progresses
directly to x_(n+1), in steps of the largest possible size
consistent with (2).
Can you explain (3) a little more?
What do you mean by "once a_1, a_2, ... hits x_n?"
More precisely, once the set {a_1, ..., a_j} contains, as a subset,
{x_1, ..., x_n} but not x_(n+1).
And by "the sequence progresses directly to x_(n+1)?"
Assume {a_1, ..., a_j} contains {x_1, ..., x_n} but not x_(n+1).
Define a_(j+1) as follows ...
If a_j < x_(n+1), let a_(j+1) = min( a_j + 1/j, x_(n+1) ).
If a_j > x_(n+1), let a_(j+1) = max( a_j - 1/j, x_(n+1) ).
In other words, define the terms a_(j+1), ... as above, as many new
terms as needed, proceeding towards x_(n+1) at the maximum speed limit
(condition (2)), until close enough to hit x_(n+1) in one step.
Note that since the harmonic series diverges, we do reach x_(n+1) in a
finite number of steps.
Also, there is no need to have the a_j terms avoid non-targeted x_k
terms -- duplicates are allowed.
Convergence of sum( (a_(n+1) - a_n)^2 ) is guaranteed by condition
(2).
Conclude that {a_1, a_2, a_3, ...} = Q and sum( (a_(n+1) - a_n)^2 )
converges.
quasi
Ah... Ok. Thanks for explaining it.
--
-kira
.
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