III ϵ-n ϵ-n lim n a n = α n a n α 1 lim a n = 0 1 n a k n n k= ϵ-n 1.1

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 1 1 1.1 ϵ-n ϵ-n lim n = α n n α 1 lim n = 0 1 n k n k=1 0 1.1.7 ϵ-n 1.1.1 n α n n α lim n = α ϵ Nϵ n > Nϵ n α < ϵ 1.1.1 ϵ n > Nϵ n α < ϵ 1.1.2 Nϵ ϵ > 0 Nϵ n > Nϵ = n α < ϵ 1.1.3 1

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 2 Nε 1 Nε 2 ε 1 ε 2 α ε ε 2 1 n Nϵ N ϵ ϵ- 1.1.3 n > Nϵ n α < ϵ n Nϵ n α ϵ Nϵ Nϵ Nϵ ϵ N ϵ N ϵ lim n = + 1.1.2 n n n lim n = + M NM n > NM n > M 1.1.4 lim n = + lim n = { n } 1.1.1 1.1.1 1 n n N n N

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 3 N N N = 10 4 N = 10 10 N = 10 100 N n n = 1/n n n n ϵ > 0 n n α ϵ ϵ ϵ ϵ = 10 6 ϵ = 10 14 ϵ = 10 200 N ϵ n α N ϵ 2 n α n α n n = 1/n ϵ = 0.0001 n > 100 n > 100 n α < 0.0001 ϵ = 10 6 n > 20000 n > 20000 n α < 10 6 ϵ = 10 12 n > 10 20 ϵ = 10 100 n > 10 300 ϵ > 0 lim n = α ϵ = 10 300 N lim n = α N ϵ ϵ-n N ϵ n n α n n α ϵ n α N n ϵ n α ϵ-n ϵ Nϵ ϵ N n = 1, 2, 3,... n = 1 n, b n = 1 log2 + log2 + log n, c 1 n = log2 + log2 + log n + 10 8 1.1.5 n n 1 10 100 10 3 10 4 10 5 10 6 10 8 10 16 n 1 10 1 10 2 10 3 10 4 10 5 10 6 10 8 10 16 b n 1.00938 0.80577 0.73645 0.69834 0.67321 0.65494 0.64084 0.62006 0.57692 c n 1.00938 0.80577 0.73645 0.69834 0.67321 0.65494 0.64084 0.62006 0.57692 2

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 4 n b n c n b n c n n n n 1/n b n log n c n 10 8 n n n N ϵ n α ϵ ϵ n n n α ϵ ϵ-n 1.1.2 1.1.7 1.1.3 n Nϵ n = 1, 2, 3,... n = 3, b n = 1 n, c n = 1, d n = 1 n n 2 + 1 1 n 10, 10 2, 10 3, 10 4, 10 5, 10 6,... e n = 0 1.1.6 1.1.7 1.1.5 n f n = n + 3 n, g n = sin n n, h n = n + 1 n, p n = 2n + 1 n + 1, q 1 n = logn + 1 1.1.8 ϵ-n 1.1.4 ϵ-n lim n = α, lim b n = β lim n + b n = α + β. lim n = α, lim b n = β lim nb n = αβ. lim n = α, lim b n n = β β 0 lim = α b n β. b n m b m = 0 {b n } 1.1.5 n = 1 + 1 n 1.1.6 1 1.1.6 n n lim n = α lim n = β α = β

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 5 ϵ-n 1.1.7 n b n = 1 n n k=1 k lim n = α lim b n = α ϵ-n 1.1.8 1.1.7 lim 1 + 2 + + n n = α = lim = α n 1 n ρ 1, ρ 2, ρ 3,... n / n b n := ρ j j ρ j j=1 lim n = α lim b n = α ρ 1, ρ 2, ρ 3,... 1.1.7 ρ 1 = ρ 2 = ρ 3 =... = 1 j=1 1.2 ϵ-δ n n x x fx 1.2.1 fx, b fx x b lim x fx = b ϵ δϵ 0 < x < δϵ x fx b < ϵ 1.2.1 ϵ > 0 δϵ > 0 0 < x < δϵ = fx b < ϵ 1.2.2 x > 0 x = fx f b f = b x

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 6 δε 2 b ε 2 ε 2 ε 1 ε 1 x δε 1 ϵ-n 0 < x < δϵ fx b < ϵ 0 < x δϵ fx b ϵ 0 < x ϵ-n ϵ, δ ϵ, δ x fx b ϵ-n ϵ δ ϵ-n α fx b < ϵ δϵ 1.2.2 δϵ 1 lim x 0 x, > 0 2 lim x 2 2x + 3 x 0, 3 lim x 2 2x + 3. 1.2.3 x 1 1 x 2 1 4 lim, 5 lim x 0 1 + x x 1 x 1, 6 lim sin 1 x 0 x, 1.2.4 x 3 3 7 lim x x 1 + x 1 x 8 lim x 0 x 9 lim x 0 x 1.2.5 1.2.3 fx lim fx x 0 ϵ-δ 0.001 x = 10 1, 10 2, 10 3, 10 4,... fx := x { } { } 1.2.4 lim fx = α lim gx = β lim fx + gx = α + β lim fxgx = αβ x x x x ϵ-δ

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 7 1.3 1.4 1.3.1 1, 2, 3,... { n } { n } { n } 1, 2, 3, 4, 5, 6,... 1, 3, 5, 7, 9,... 1, 4, 9, 16, 25,... 1, 2, 5, 10, 100, 10032, 2323445,... 1.3.2 { n } L n n < L K n n > K K, L { n } n n L n K 1.3.3 { n } {b n } {b n } 1, 2, 3,... K L ccumultion point 11 23 K 1 4 2 3 5 8 15 9 100 12 L

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 8 n 2 n 1.3.1 1 = 1.4, 2 = 1.41, 3 = 1.414,... 2 II Bolzno-Weiertrss 1.4 lim n = α n α ϵ > 0 Nϵ n > Nϵ n α < ϵ 1.4.1 α e e = lim 1 + 1 n 1.4.2 n e x = 1 + x + x2 2! + x3 3! + = lim N N n=0 x n n! 1.4.3 x e x e x 3 lim N N n=0 x n n n! lim N N n=0 x n n n! 1.4.4 3 e x

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 9 1.4.1 1 2 3... n... n monotone incresing monotone decresing monotone non-decresing monotone non-incresing. strictly incresing n n 1.4.2 { n } lim n { n } lim n { n } lim n = + { n } lim n = + ± lim n 1.4.2 n 2 n n 2

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 10 n 1.4.2 n ϵ-δ α 1.3.3 { n } {b k } α { n } α α {b k } α k b k α 1.4.5 {b k } { n} {b k } k 1 b k1 > α n 1 k b k b k1 > α b k α {b k } { n } k n b k = n 1.4.5 n = b k n n α n n n α n m m n α {b k } k n = b k n n n α 1.4.6 { n }, {b k } {b n } α ϵ > 0 Kϵ > 0 k > Kϵ = b k α < ϵ 1.4.7 k > Kϵ α ϵ < b k 1.4.8 n = b k n α ϵ < n { n } n 1 α ϵ < n1 n > n 1 α ϵ < n1 n ϵ > 0 1.4.7 Kϵ Kϵ k 1 n1 = b k1 n 1 n > n 1 α ϵ < n 1.4.9

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 11 1.4.6 ϵ > 0 n 1 > 0 n > n 1 α ϵ < n < α 1.4.10 lim n = α ϵ-δ { n } α α 1.5 1.5.1 n Cuchy sequence ϵ > 0 Nϵ m, n Nϵ m n < ϵ 1.5.1 ε 1 ε 2 Nε 1 Nε 2 n n n m m, n 1.5.2 n n lim sup lim inf

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 12 1.5.3 1.5.2 Nϵ n := 1 n b n := 1 n 2 c n := 1n n d n := 1n n 1.5.4 { n }, {b n }, {c n } α c n n := log n + n k=1 1 k b n := c 1 := 1, n 1 c n+1 := 1 2 n 1 k 1 k=1 k c n + α c n c n n, b n 1.5.5 1.4.4 e x e x = n=0 x n n! x x > 0 x sin x = x x3 3! + x5 5! x7 7! + x 0 < r < 1 { n } n+2 n+1 r n+1 n n = 1, 2, 3,... n x 1.5.6 lim x fx fx C ϵ > 0 δϵ > 0 0 < x < δϵ 0 < y < δϵ x, y fx fy < ϵ 1.6

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 13 1.6.1 A N A N A bounded from bove N A upper bound M A M A bounded from below M A lower bound A A bounded A [0, 1] 1 10 2345 A 1 123 33556 A A A 1.6.2 A A A A supremum sup A A A A infimum inf A A A A inf A sup A A A 1.6.3 1.6.3 S S [ ] [ ] S S [ ] n 1.6.4 { n } lim sup k k n 1.6.1 { n } { n } lim sup n lim n 1.6.2 lim inf k k n 1.6.3 lim inf n lim n 1.6.4 sup k n k n +

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 14 α α α 1.6.5 { n } n α ϵ > 0 Nϵ/2 n > Nϵ/2 n α < ϵ/2 m, n > Nϵ/2 m α < ϵ 2, n α < ϵ 2 1.6.5 m, n m n = m α + α n m α + α n < ϵ 2 + ϵ 2 = ϵ 1.6.6 1.5.1 β := lim inf n, γ := lim sup n 1.6.7 β γ b N := inf m N m, c N := sup n 1.6.8 n N 1.6.5 β = γ { n } ϵ > 0, N l, m N = l m < ϵ 1.6.9 ϵ > 0, N l, m N = l m < ϵ 1.6.10 N, m l N sup sup l = c N l N ϵ > 0, N m N = c N m ϵ 1.6.11

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 15 m N inf inf m N l = b N ϵ > 0, N c N b N ϵ 1.6.12 c n b n {b n } {c n } {c n b n } N c N b N ϵ n N c n b n ϵ 1.6.12 ϵ > 0, N n N = c n b n ϵ 1.6.13 lim c n b n = 0 ϵ-n β = γ

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 16 2 2.1 2.1.1 [, b] f n x n = 1, 2, 3,... i {f n x} [, b] fx x lim f nx = fx ϵ > 0 x [, b] Nϵ, x n > Nϵ, x = f n x fx < ϵ 2.1.1 ii {f n x} [, b] fx ϵ > 0 Nϵ x [, b] n > Nϵ = f n x fx < ϵ 2.1.2 lim f nx = fx N x x N N x I = [, b] f n x n = 1, 2, 3,... 2.2 f n x n 0 < x < 1/n f n x = 0 2.2.1 [ 1 ] lim f n xdx = 0 1 0 lim f nx dx 2.2.2 4 lim 1 0 1 0 < x < 1/n g n x = 2.2.3 0 4

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 17 [ 1 ] lim g n xdx = 0 fx = lim f nx 2.2.2 1 [ 1 ] lim f n xdx = 0 lim [ 1 0 0 lim g nx dx 2.2.4 1 0 fxdx 2.2.5 { } ] f n x fx dx = 0 2.2.6 g n x = f n x fx [, b] g n x x lim g nx = 0 lim g n x = 0 lim g nx = 0 ϵ > 0 Nϵ, x n > Nϵ, x = g n x < ϵ 2.2.7 x g n x < ϵ g n xdx g n x dx b ϵ 2.2.8 lim g nx = 0 2.2.7 N x x g n x < ϵ n 2.2.1 f n x g n x = f n x 0 f n x n n x g n x < ϵ 2.2.8 2.2.1 [, b] f n x n = 1, 2, 3,... fx lim f n xdx = { } lim f nx dx = fxdx 2.2.9 2.2.9 2.2.9 α f n x fx f fx

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 18 ϵ > 0 Nϵ x [, b] n > Nϵ = f n x fx < ϵ 2.2.10 ϵ f n xdx f n xdx fxdx = fxdx f n x fx dx {f n x fx}dx 2.2.11 ϵ dx = ϵb 2.2.12 ϵ n n n f n x 2.2.2 i I = [, b] n=0 f nx F x f n x [ n=0 ] f n xdx = [ n=0 ] f n x dx = F xdx. iii n=0 nx n R fx n=0 nx n R, R [, b] [ ] fxdx = n x n dx = 0, b = x x < R x 0 ftdt = n=0 n=0 n n + 1 xn+1. 2.2.3 Arzelà 5.10, 5.11 I =, b {f n x} n n, x M n 0 x I fx M fx := lim f nx I dx n lim fxdx = lim f nx dx = fxdx 2.2.13 2.2.4 5.12

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 19 I =, {f n x} gx g f n n 0 x I f n x gx gxdx < fx := lim f nx I dx n lim fxdx = lim f nx dx = fxdx 2.2.14 5.4 2.2.4 2.3 I = [, b] f n x n = 1, 2, 3,... x I lim f nx = fx n f n x x fx x [ 1, 1] 0 1 x 0 f n x = nx 0 < x 1/n 1 1/n < x 1 2.3.1 n fx = lim f 0 1 x 0 nx = 1 0 < x 1 2.3.2 x = 0 2.3.1 I {f n } fx f n x fx

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 20 fx I = [, b] c lim fx = fc, ϵ > 0, δ > 0 x c < δ = fx fc < ϵ 2.3.3 x c f n x fx fc f n x f n c fx fc = fx f n x + f n x f n c + f n c fc 2.3.4 fx fc fx f n x + f n x f n c + f n c fc 2.3.5 f n x fx f n c fc f n f n f n x f n c f n x c x c n 2.3.5 c n x n x n x c n n x c n 2.3.1 c = 0 2.3.5 n x > 0 0 1 0 = 1 x > 0 n 1 0 = 1 x c n ϵ > 0 2.3.5 ϵ Nϵ x n Nϵ ϵ n Nϵ 2.3.5 ϵ x f Nϵ x δϵ, Nϵ > 0 x c < δϵ, Nϵ 2.3.5 ϵ ϵ, Nϵ, δϵ, Nϵ 2.3.5 ϵ 2.3.5 3ϵ ϵ x c fx fc < 3ϵ fx x = c 2.4 f n x fx 2.4.1 i f n x I = [, b] C 1 - { d dx f nx} I {f n x} x 0 I {f n x} I C 1 - ii f n x I C 1 - n [ ] d lim dx f nx = d [ ] lim dx f nx = d dx fx. d dx f nx I n f nx

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 21 x 0 I n f nx I C 1 - [ ] d dx f nx n=0 = d [ ] f n x. dx n=0 iii n nx n R fx n nx n R, R d dx fx = n n x n 1 n=1 2.5 R {x, y x [, b], y [c, d]} fx, y fx, y x y Iy y Iy fx, ydx. 2.5.1 fx, y R y [c, d] Iy R fx, y y [c, d] Iy y y d dy Iy = d [ ] fx, ydx = dy [ ] fx, y dx y 2.6 fx = n x n f x = n nx n 1?? 2.6.1 n=0 n=1 2.2.1 2.2.2 2.4.1 Newton, Leibnitz x

III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 22