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[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

[72] Basic arithmetic properties of divergent functions

https://www.evernote.com/shard/s29/sh/ea9f00a6-dea1-44f9-ab2a-1ae65c60798a/e1da116caa4b5793e08b33628f8a176e

[71] f(x) → b (x → a). ⇔ f_i(x) → b_i (x → a) for all 1 ≦ i ≦ m, provided that f(x) = (f_1(x), . . . f_m(x)) and b = (b_1, . . . b_m).

https://www.evernote.com/shard/s29/sh/d691b66b-6f42-41a5-8036-7dc14212f8b0/059b33aa8a475513edb54630e0f306da

[70] √x is continuous in [0, +∞).

https://www.evernote.com/shard/s29/sh/dcb4b844-c3db-493e-8e16-111b2f8083ab/1e4a0921b9e9df44c97e07724182b9d7

[69] f(x) → b (x → a) & g(y) → c (y → b) ⇒ (g・f)(x) → c (x → a).

https://www.evernote.com/shard/s29/sh/481aceb1-e26e-4fd4-8729-5ac088f0f122/c44318ab25278e31351ffb8490a2b45e

[68] If f and g are continuous at a, then (f±g), kf, fg, and f/g (g(a) ≠ 0) are also continuous at a.

https://www.evernote.com/shard/s29/sh/3fcdc42d-51c7-4a10-bcc6-fc92934cccab/e6961d8bef7b69e03322719b33afe0e9

[67] lim (f(x)±g(x)) = lim f(x) ± lim g(x), lim kf(x) = k・lim f(x)

https://www.evernote.com/shard/s29/sh/e1df7638-806c-4c0a-82dc-546accd11a3f/74aae0c20585ba39a252273f05de493f

[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

[65] If lim f(x) = b and lim g(x) = c, then lim f(x)・g(x) = bc, and lim f(x)/g(x) = b/c (if c≠0)

https://www.evernote.com/shard/s29/sh/e64a8d45-e3e6-43d4-acf9-33cb06bde025/14b2da255367f352d204c48031c58900

[64] f(x)→b (x→a) ⇔ ∀ε＞0. ∃δ＞0. f(U(a,δ)∩D) ⊂ U(b,ε) ／ ∃lim f(x) (x→a) ⇒ f is bounded in some neighborhood of a in D.

https://www.evernote.com/shard/s29/sh/9860ca20-334f-4ace-a06b-18088739686a/d946b21aa1c8c0a051d32350191171ec

[63] f is continuous at a ⇔ . . . （関数fは点aにおいて連続である）

https://www.evernote.com/shard/s29/sh/1328340f-e265-4a4b-ba89-6c75fde634dd/83f6cf4e58ff752b66372941b56cb591

[62] ∃limf(x) ⇒ ∃!limf(x).

https://www.evernote.com/shard/s29/sh/c96ef008-7316-4a37-a263-cda7ef498811/7b92cac8e8cec3343adf7566a2177e5d

[61] If B⊂A⊂R^n, f:A→R^m, and a ∈ the closure of B, then: f(x)→b (x→a, x∈B). ⇔ f(x_n)→b (n→∞) for any sequence (x_n) of points in B s.t. x_n→a (n→∞).

https://www.evernote.com/shard/s29/sh/5ba9c949-2df2-4904-849f-dcae40cccafd/596e2bb369518a726be0f5814d9e3b83

[A2] Axiom of choice (選択公理): For every family {A_λ≠∅}λ∈Λ of nonempty sets, Π_{λ∈Λ} A_λ ≠ ∅

[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

[58] For a nonnegative term series Σa_n, if there exists l ∈ R ∪ {+∞} s.t. a_{n+1}/a_n → l (n → ∞), then Σa_n converges if l ＜ 1, and Σa_n diverges if l ＞ 1.

https://www.evernote.com/shard/s29/sh/6d93b03e-5842-4301-bf2e-64574de00c33/3ead698a49bac72999dcc8d6f81a27dc

[57] For any nonnegative term series Σa_n, [0≦∃k＜1. ∃m∈N. ∀n∈N. n≧m ⇒ (n√a_n ≦ k ∨ a_{n+1}/a_n ≦ k)] ⇒ Σa_n converges.

https://www.evernote.com/shard/s29/sh/b5fc9d77-98ad-415e-bf5a-c12b66af0577/7442e7245e0720d417c5c0c75ed99748

[56] If z∈C and |z| > 1, then Σz^n diverges.

https://www.evernote.com/shard/s29/sh/46d80a96-1cb9-42ef-95dc-220f1306b0f5/65037777da64feabad96e524519b84f6

[55] If z∈C & |z|＜1, then Σz^n converges to 1/1-z.

https://www.evernote.com/shard/s29/sh/d472bffb-4b6e-42d1-b1d3-fb75f1aee008/cdc8968f534313a7d0cdf695a91db091

[54] 0≦x≦1 ⇒ x^n → 0 (n → ∞)

https://www.evernote.com/shard/s29/sh/96b21d75-e2d1-49fe-8a14-b62ac4bf9127/293e03f0947b7a338d606efe5378208c

[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

[44] For any x,y ∈ R^n, a ∈ R, |x+y| ≦ |x| + |y| (the equality holds only when x and y are linearly dependent and (x|y) ≧ 0.)

https://www.evernote.com/shard/s29/sh/86a5b991-db49-43a5-b524-d246fc8c3af7/80bba97621801cda77158fdaeb1bc848

[43] Schwarz Inequality: For any x,y∈R^n, |(x|y)| ≦ |x||y| (the equality holds only if tx = y for some t∈R.)

https://www.evernote.com/shard/s29/sh/689a6187-d66b-4e6c-8fd8-d681267fd5f0/5a90346efdf827ddbc6ad9ab00d2a663

[42] For any x,y ∈ R^n, |x±y|^2 = |x|^2 ± 2(x|y) + |y|^2

https://www.evernote.com/shard/s29/sh/ecc25315-76e9-49c4-9df0-30b6685c89c5/dac373916e58b293996f9912ac0cb1ea

[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

[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