>>447
>>任意実数は 有理数からなる コーシー列の収束先として定義できる(>>331)
>なんてアホなこと言わないはずだぞ

何を言いたいのか 意味が分らんぞｗ
有理数 → 有限小数 と言い換えて
「任意実数は 有限小数からなる コーシー列の収束先として定義できる」
と言ったら なんか 文句ある？ ｗ

有限小数 ⊂ 有理数
だよね

例えば
円周率 π＝3.14159265・・・
つまり
3→3.1→3.14→3.141→3.1415→3.14159→3.141592→3.1415926→3.1415265→・・・
と 小数点以下を一桁ずつ 増やす数列で π に収束する 数列が作れるよね
そう 言ったら 文句あるか？ ｗ

そして、この数列で 有限小数 → 無限小数（有限小数の極限）だと考えたのがカントールさん
カントールは、これで 対角線論法を 考え出したことは 有名だね（下記 en.wikipedia ご参照）
但し、10進小数でなく 2進小数だったそうな

(参考)
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
Cantor's diagonal argument

https://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Diagonal_argument_01_svg.svg/375px-Diagonal_argument_01_svg.svg.png
An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above.

Uncountable set
Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one).[note 2] He begins with a constructive proof of the following lemma:

If s1, s2, ... , sn, ... is any enumeration of elements from T,[note 3] then an element s of T can be constructed that doesn't correspond to any sn in the enumeration.
The proof starts with an enumeration of elements from T, for example
s1 =(0,0,0,0,0,0,0,...)
s2 =(1,1,1,1,1,1,1,...)
s3 =(0,1,0,1,0,1,0,...)
s4 =(1,0,1,0,1,0,1,...)
s5 =(1,1,0,1,0,1,1,...)
s6 =(0,0,1,1,0,1,1,...)
s7 =(1,0,0,0,1,0,0,...)
...

Next, a sequence s is constructed by choosing the 1st digit as complementary to the 1st digit of s1 (swapping 0s for 1s and vice versa), the 2nd digit as complementary to the 2nd digit of s2, the 3rd digit as complementary to the 3rd digit of s3, and generally for every n, the n-th digit as complementary to the n-th digit of sn. For the example above, this yields

s1=(0,0,0,0,0,0,0,...)
s2=(1,1,1,1,1,1,1,...)
s3=(0,1,0,1,0,1,0,...)
s4=(1,0,1,0,1,0,1,...)
s5=(1,1,0,1,0,1,1,...)
s6=(0,0,1,1,0,1,1,...)
s7=(1,0,0,0,1,0,0,...)
...
s=(1,0,1,1,1,0,1,...)
By construction, s is a member of T that differs from each sn, since their n-th digits differ (highlighted in the example). Hence, s cannot occur in the enumeration.