[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

[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

[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

[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