Is this proof valid?

kindlychung

[这个贴子最后由kindlychung在 2010/09/25 02:14am 第 1 次编辑]

This is a proof for the uncountability of R, yet it';s not the well-known diagonalization method. Is the application of Nested Interval Property legitimate?

elimqiu

Sure it is.

kindlychung

But it seems NIP only applies when I_n are closed.
In this proof, I_n are not even bounded on the right.

elimqiu

I_1 is bounded, closed, and so are I_n according to the way of construction

kindlychung

I see, I misunderstood it. In fact, there is a much simpler proof by density of R. If R can be enumerated in a list, then between any two real numbers a,b in the list, another real c cannot be found such that a

elimqiu

下面引用由kindlychung在 2010/09/26 01:15am 发表的内容：
I see, I misunderstood it.
In fact, there is a much simpler proof by density of R.
If R can be enumerated in a list, then between any two real numbers a,b in the list, another real c cannot be fou ...

This is wrong. All rationals got the property but they';re countable.

elimqiu

Is the your official textbook or you just learn from yourself?
If the latter, how did you decide use this one? Just curious

kindlychung

I teach myself, and I pick my own textbooks.
The title of this book is attractive, the content is rigorous and detailed, I like it. And if one wants to understand something, one has to start somewhere.
[br][br]-=-=-=-=- 以下内容由 kindlychung 在 时添加 -=-=-=-=-
Do you have any recommendations?

kindlychung

[这个贴子最后由kindlychung在 2010/09/26 03:33pm 第 1 次编辑] Then, for any two rational numbers q1

elimqiu

下面引用由kindlychung在 2010/09/26 02:14pm 发表的内容： Then, for any two rational numbers q1

card ((q1,q2)∩Q) = card Q[br][br]-=-=-=-=- 以下内容由 elimqiu 在 时添加 -=-=-=-=- The book is good for self study.