Is it possible to construct a sequence like this?

kindlychung

Is it possible to construct a sequence like this? If yes, please construct one, if not, give a proof.

elimqiu

If there is such a sequence, then 0 must also be the limit of some sub-sequences of it. So there is no sequence with the given set as its total limits of sub-sequences.
On the other hand, if you list all the rationals in (0,1), then the resulting sequence makes all the whole [0,1] as the set of the limits of its sub-sequences.
Moreover I can cocnstruct a sequence such that { x | x = 0 or x = 1/m, m is positive integer}  is exactly the total limits of its sub-sequences. But I';m not sure you want to know it. It';s a little complicated.[br][br]-=-=-=-=- 以下内容由 elimqiu 在 时添加 -=-=-=-=-
If we take the 1-1 correspondence f:N →N×N where N is the set of positive integers,
Denote f(n) = (p(n),q(n)), then
{ 1/p(n) + 1/(p(n)q(n)) } should be such a sequence.
Of course, there are many solutions to the problem.

kindlychung

Moreover I can cocnstruct a sequence such that { x | x = 0 or x = 1/m, m is positive integer}  is exactly the total limits of its sub-sequences. But I';m not sure you want to know it. It';s a little complicated.

Please post it, I';m interested.

elimqiu

This one
http://www.mathchina.com/cgi-bin/topic.cgi?forum=5&topic=10176
Actually tells you how to construct a function f = (f1,f2) from N to N^2.
Now { 1/f1(n) + 1/f2(n) } is what you want.

elimqiu

Actually here is the much intuitive one:

1,1/2,1,1/2,1/3,1,1/2,1/3,1/4,1,1/2,1/3,1/4,1/5,....., 1,1/2,...1/n, 1,1/2,...,1/(n+1), .....

kindlychung

Yes, very intuitive. Thanks.