[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