[0041] 10進数（又はn進数）小数展開の存在／The existence of decimal (or n-ary for any n∈N) representation of any real numbers

https://www.evernote.com/shard/s29/sh/9849a0ba-a5ce-404e-8a22-24cbf2532c32/238dc3d467b3f671c0448c7b275d8ed9

[40] ∀a,b∈R (a＜b). ∃x∈Q. a ＜ x ＜ b.

https://www.evernote.com/shard/s29/sh/f20363cf-8035-4086-b610-21dbc89a87aa/290b8a75a7eadd104a5f4057e48e41b4

[39] The unique existence of ⎿x⏌(the floor of x): ∀a∈R. ∃!n∈Z. n ≦ a ＜ n+1

https://www.evernote.com/shard/s29/sh/9b731c36-15ae-4a58-b6c0-16865eb87482/489a6181cdcf6cd2359b7864bc373c61

[34] [The unique existence of the mth root／m乗根の存在の一意性] ∀m≧2∈N. ∀a＞0∈R. ∃!b＞0∈R. b^m = a.

https://www.evernote.com/shard/s29/sh/06250865-dca2-4db9-9c0b-db776543944c/f3dc2631a8ae7098df443239d0f3e8eb 

[0032] 1/n → 0 (as n → ∞)

https://www.evernote.com/shard/s29/sh/8eb4c32e-0d4e-468c-b021-d116822dcae6/27f4d3221cfa469535e3eb9415c24c4e

[31] a>0 ⇒ na → +∞ (as n → ∞) (⇔[25] Archimedean Principle)

http://www.evernote.com/shard/s29/sh/c9197384-e140-4f42-9c6a-a1c19e41e308/be0b8575b4e951c053468554330c353c

[28] For an ordered field K, ∀a,b∈K. a≦b ∧ c≧0 ⇒ ac≦bc, a≦b ∧ c≦0 ⇒ ac≧bc

http://www.evernote.com/shard/s29/sh/07b04633-3473-4674-a00c-dfe41ec62aeb/e6746a682987953882c5bf12a64b4e0b

[27] [25] ∀a,b>0∈R. ∃n∈N. na>b ⇔ [26] ∀c∈R. ∃n∈N.

http://www.evernote.com/shard/s29/sh/435cd3a1-43a6-4931-8c84-18851d3e8240/7f3521d135469845a5460be764abd04f

[26] ∀a∈R. ∃n∈N. a＜n ⇔ N is not a subset of R bounded from above.

http://t.co/dG0QxXx

[A1] 17 axioms for real numbers／実数に関する17の公理

http://t.co/ZCayUrS

[25] Archimedean Principle／アルキメデスの原理: ∀a,b＞0 ∈ R. ∃n∈N. na ＞ b.

http://t.co/5MQzcjh

[12] x^n - y^n = (x - y) (x^{n-1} + x^{n-2}・y +…+ x・y^{n-2} + y^{n-1})

http://t.co/JhUGD2M

[7] Any positive integer greater than 1 can be written as a unique product of prime numbers.／素因数分解の一意性

http://t.co/g4Qh6p7

[5] If p is a natural number, then the nth root of p (n√p) is not a rational number.／自然数pのn乗根は無理数である

To be modified.

[2] Infinity of prime numbers／素数は無限に存在する

http://t.co/3oLuGEj