[38] Cauchy's Convergence Criterion: A sequence of real numbers is convergent iff it is a Cauchy sequence. ／コーシーの収束条件: 実数列の収束の必要十分条件はそれがコーシー列となることである。

https://www.evernote.com/shard/s29/sh/a4c0ec2c-f326-4417-9014-5418045dde56/4b9e7bf8b9a0d4190c89d55f5bda47d9

[37] If a subsequence {a_n(k)} of a Cauchy sequence {a_n} converges to a, {a_n} converges to a. ／ コーシー列の部分列がaに収束するならば、コーシー列自体もaに収束する。

https://www.evernote.com/shard/s29/sh/3f353348-a241-4090-859f-2c4dcbe441d9/9afdc87d51f57956b37b4598439f99d6

[36] Every Cauchy sequence is bounded. ／ コーシー列は有界である。

https://www.evernote.com/shard/s29/sh/b54545ea-0690-48bf-b032-087b68db573c/b103cafcd1e6c8b2e191f17738c1c201

[35] The Bolzano-Weierstrass theorem Every bounded sequence has a convergent subsequence.／ボルツァーノ・ワイエルシュトラスの定理 有界列は収束する部分列を持つ。

https://www.evernote.com/shard/s29/sh/664daec1-f9f7-4cbf-a095-669f4a20a3da/c0079223080bbd3cf3b35d606375ea1e

[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 

[33] 2^n → +∞, 1/2^n → 0 (as n → ∞)

https://www.evernote.com/shard/s29/sh/c4e46234-9514-4b2e-9aae-1ca14cf90aef/5177a887660a1f2a72ee2d016080e81e

[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

[30] [Nested interval theorem／区間縮小法(2)] ∀n∈N. [a_n, b_n] ⊃ [a_n+1, b_n+1] ∧ b_n - a_n → 0 (as n → ∞) ⇒ ∃c∈R. ∩_n∈N [a_n, b_n] = {c} ∧ a_n, b_n → c (as n → ∞)

http://www.evernote.com/shard/s29/sh/62196813-6786-497f-b308-2e8a7efe2cf5/c032cc13c1673d3a4ea2cd73db15a6f2

[29] [Nested interval theorem／区間縮小法(1)] For a sequence of closed bounded intervals {I_n} s.t. I_n ⊃ I_n+1, there is a real number that belongs to all I_n.

http://www.evernote.com/shard/s29/sh/6e8ba429-4fc1-40bd-b924-2e4336cc8a76/c1fa72632d4dcf9c811bb67f1187f83b

[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

[24] If {a_n} is monotonically increasing/decreasing and bounded from above/below, it converges to sup/inf {a_n}／上/下に有界な単調増加実数列{a_n}はその上限/下限に収束する

http://t.co/pGuqFZW

[23] a_n → a (as n → ∞) ∧ b_n → b (as n → ∞) ⇒ a_n・b_n → ab (as n → ∞)

http://t.co/qnqYwLk

[22] a_n → a ≠ 0 (n → ∞) ⇒ 1/a_n → 1/a (n → ∞)

http://t.co/TgtejCc

[21] |a + b| ≦ |a| + |b| & |a| - |b| ≦ |a ± b|

http://t.co/okQSv2N

[20] {f(g(x))}’ = f’(g(x))・g’(x)

http://t.co/WWH1bWi

[19] {f(x)・g(x)}’ = f’(x)・g(x) + f(x)・g’(x)

http://t.co/oIsSosQ

[18] x + y + z = π (half circle) & x,y,z ≠ π/2 ⇒ tanx + tany + tanz = tanx・tany・tanz

http://t.co/euqynBA

http://t.co/fVTSnSx

[17] cos’θ = -sinθ

http://t.co/sfn77Lx

[16] sin’θ = cosθ

http://t.co/CZNbiJ5