>>313 補足
> 2．”Without Choice, you may have Dedekind reals that do not correspond to any sequence of Cauchy reals.”
>　(google訳 選択公理がない場合、コーシー実数のどの列にも対応しないデデキント実数が存在する可能性があります）
>　思うに、選択公理がない場合は コーシー実数の構成が不十分で 不足する部分があるよと

ここを補足しておくよ （＾＾

https://en.wikipedia.org/wiki/Dedekind-infinite_set
Dedekind-infinite set
References
・Jech, Thomas J., The Axiom of Choice, Dover Publications, 2008, ISBN 0-486-46624-8
　↓
これから 検索で 1973年版がヒットした
CH.2, THE COUNTABLE AXIOM OF CHOICE
より 2.4.3. Example が参考になるだろう
（下記 画像urlと pdfからのテキスト部分のコピーを貼る）

https://imgur.com/a/6QTdkX5
Jech, T The Axiom of Choice. North-Holland. Jelonek 1973
CH.2, THE COUNTABLE AXIOM OF CHOICE
（ちょっと文字化け ご容赦 上記のURL画像ご参照）
P21
2.4.3. Example: Topological properties of the real line.
There are two kinds of definitions of the basic topological properties of
the real line: (1) &-&definitions and (2) definitions using limits of
sequences.
(a). Closed sets: A point x is in the closure of a set A if
(1) every neighborhood of x intersects A;
(2) x = 1imnAmx,, for some sequence {x,} of points in A.

(b). Continuous functions: A function f is continuous at a point x if
(1) V~36etc.
(2) whenever lim x, = x, then lim f(xJ = f(x).

(c). Compact sets: A set A is compact if
(1) A is dosed and bounded;
(2) every sequence {x,} of points in A has a convergent subsequence

(Notice that the Heine-Bore1 theorem can be proved for the closed and
bounded sets - see Problem 25).
PROPOSITION. In the presence of the Countable Axiom of Choice, the de-
$nitdons under (1) and (2) are equivalent in each of the cases (a), (b), (c).
PROOF. One direction is trivial in each case.
(a). If x is in the closure of A, then each (l/n)-neighborhood of x inter-
sects A. We can choose x, by the Countable Axiom of Choice.
(b). Iff is discontinuous at x, then there is E > 0 such that for each n,
we can choose x, in the (l/n)-neighborhood of x such that If (x,) - f (x)l 2 E.
Then f (x) = lim f (x,) is false.
with lim x, E A.
(c). Similar.