[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