[73] If there exists b = lim f(x_n) for an arbitrary sequence (x_n| x_n → a, n → ∞), then b is unique for all (x_n).

https://www.evernote.com/shard/s29/sh/cd85a330-27a7-48ef-bf8d-112d0a80acf5/1b5a733aead4e22bfb97a5f2d5cdc530

[66] An arbitrary subsequence converges to a ⇒ The original sequence also converges to a.

https://www.evernote.com/shard/s29/sh/c6c652d7-537a-4f44-b9bb-595e06ac8ffb/f472ef9a3cfec9b8694e2f8d351ace0d

[60] For any A ⊂ R^n, b ∈ A. ⇔ There exists a sequence of points in A that converges to b.

https://www.evernote.com/shard/s29/sh/b3bf6ce2-2f62-4b9c-9b0d-63a0b4e88deb/6c2b92dddab0fef9f2c1af6674a603b5

[59] (a_n) converges to a ⇒ Any subsequence of (a_n) also converges to a.

https://www.evernote.com/shard/s29/sh/7cfe1d8c-0c23-4a85-bf2b-2875aac0f407/06684f4b4e8adb179589f3e983027bdd

[53] Limit comparison theorem／極限に関する比較定理

https://www.evernote.com/shard/s29/sh/70ff849c-7247-493d-8c3c-ea759405624e/c44fe7b65e87788ff92a66ce01faaff5

[52] A nonnegative term series Σa_n converges ⇔ The sequence of partial sums (S_n)n∈N is bounded above.

https://www.evernote.com/shard/s29/sh/4863f174-7629-43d8-be17-672a59ba6ced/c76ada5da43dd99b3bf888581ebf0d15

[51] If Σa_n converges, then a_n → 0 (n → ∞)

https://www.evernote.com/shard/s29/sh/cac04f48-e0ef-4328-ab15-d7e7b21e3a8e/b5a800b9a9bbedea37837f83f08969aa

[50] Cauchy's Convergence Criterion on Series: A sequence of partial sums is convergent iff it is Cauchy.

https://www.evernote.com/shard/s29/sh/778c319e-040b-4bea-b21c-25af5b3c5cf8/00a97e33a2f30933fed6abb42d8ad157

[49] A sequence of points in R^n converges ⇔ It is a Cauchy sequence.

https://www.evernote.com/shard/s29/sh/f09e5c90-c804-4f47-8139-57cdc1d4549f/6c8ee2cae62e18d71a885348b9ba0923

[48] Geometric series／等比数列の和 Σa_k = a(1-r^n) / 1-r (when r≠1)

https://www.evernote.com/shard/s29/sh/f2c30088-c485-424e-8502-fdb56c95c1cf/4d51d96ffbca186c2401ea9966d4b1bf

[47] A sequence of points in R^n (a_m)m∈N converges. ⇒ (a_m)m∈N is bounded.

https://www.evernote.com/shard/s29/sh/20df19ac-ae38-4a8c-9366-1dd515518bb6/17cd6049725c2caf83c05c01da57955e

[45] A sequence of points (a_m)m∈N in R^n is Cauchy ⇔ The sequence of real numbers (a_{m,k})m∈N is Cauchy for 1≦∀k≦n.

https://www.evernote.com/shard/s29/sh/3c65d684-013d-41f8-9aca-98faf68ac62f/5406743f7d4ec6fd7354b80997113ed5

[45] A sequence (a_m) of points in R^n converges to b = (b_1, b_2,..., b_n) ⇔ 1≦∀k≦n, (a_{m,k})m∈N converges to b_k.

https://www.evernote.com/shard/s29/sh/6b698804-7ddf-4cee-909d-64234c384caa/0af43d636e71745c6edb25d992fb26db

[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 

[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