n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m

1 1 1 + 1 4 + + 1 n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m a n < ε 1

1. ε = 10 1 N m, n N a m a n < ε = 10 1 N a m, a n 1 9 1 2. ε = 10 2 N m, n N a m a n < ε = 10 2 N a m, a n 2 9 1 3. ε = 10 3 N m, n N a m a n < ε = 10 3 N a m, a n 3 9 1 4. ε 10 n n 1.3. 1. r a n = r, r = 1, 2, {a n } n=1 2. ε > 0 N N Nε > 0 n > m N a n = 1 1 2 + 1 2 2 + + 1, n = 1, 2, n2 a n a m = < 1 (m + 1) 2 + 1 (m + 2) 2 + + 1 n 2 1 m(m + 1) + + 1 n(n + 1) = 1 n 1 m + 1 < 1 n < 1 N 2

ε > 0 N N Nε > 1 n > m N a n a m = a n a m < ε n, m N a n a m < ε 1.4 ( ). {a n } n=1 Q 1. ε > 0 N N m, n N a m a n < ε (1) 2. ε > 0 N N m, n > N a m a n < ε (2) 3. ε > 0 N N m, n N a m a n ε (3) 4. ε > 0 N N m, n > N a m a n ε (4) 4. () (1) = (2), (3) = (4) (4) (1) (1) ε > 0 (4) ε > 0 1 ε (4) N N 2 n, m > N a n a m 1 2 ε N M 3

M N n, m > M a n a m 1 2 ε (1) N = M + 1 n N n > M 1 2 ε < ε n, m N a n a m < ε 1.5 ( N, ε ). {a n } n=1 Q 1. ε > 0 N N m, n N a m a n < ε (5) 2. ε > 0 N N m, n N a m a n < 2ε (6) 3. ε > 0 N N m, n N + 1 a m a n < ε (7) 4. ε > 0 N N m, n N a m a n < 1 2 ε (8) 4 4

N, ε. (3) (7) (1) (5) (6) (8) (1) (8) = (4) (1) = (6) (6) (8) (8) ε > 0 (6) 1 ε (6) N 4 (8) m, n N a m a n < 2 1 4 ε {a n } n=1 α {a n + 10 n } n=1 α 1.6 (). {a n } n=1 {b n} n=1 [ ] ε > 0 N N n N a n b n < ε 3 1.7. {a n } n=1, {b n} n=1, {c n} n=1 1. {a n } n=1 {a n} n=1 2. {a n } n=1 {b n} n=1 {b n} n=1 {a n} n=1 3. {a n } n=1 {b n} n=1 {b n} n=1 {c n} n=1 {a n } n=1 {c n} n=1. 1 2 3 {a n } n=1 {b n} n=1 {b n } n=1 {c n} n=1 [ ] ε > 0 N N n N a n b n < ε 5

[ ] ε > 0 N N n N b n c n < ε ε > 0 [ ] N 1 N n N 1 a n b n < 1 2 ε [ ] N 2 N n N 2 b n c n < 1 2 ε N = max(n 1, N 2 ) n N a n b n < 1 2 ε, b n c n < 1 2 ε α + β α + β n N a n c n a n b n + b n c n < 1 2 ε + 1 2 ε = ε 1.1 1.8 (). {a n } n=1 M a n M, n N 1.9.. {a n } n=1 ε = 1 > 0 N m, n N a n a m 1 a n M, n N M = max( a 1, a 2,, a N 1, a N + 1) 1.2 1.10. {a n } n=1, {b n} n=1 1. {a n + b n } n=1 6

2. {a n b n } n=1 3. {a n b n } n=1 4. {b n } n=1 0 ε 0 N n N b n > 1 2 ε 0 > 0 5. {b n } n=1 0 b n 0, n = 1, 2, {a n /b n } n=1 4,5 4. {b n } n=1 0 ε 0 > 0 N n N b n > ε 0 {b n } n=1 N n, m N b n b m > 1 2 ε N n N b n > ε 0 N = N n N b n > ε 0 m N b m b n b m b n > ε 0 1 2 ε 0 = 1 2 ε 0 > 0 ε 0 N n N b n > 1 2 ε 0 > 0 5. {a n /b n } n=1 M > 0 a n, b n M 4 ε 0 N n N b n > 1 2 ε 0 > 0 b 1, b 2,, b N 1 0 ( κ = min b 1, b 2,, b N 1, 1 ) 2 ε 0 b n κ > 0, n N m, n N a n a m b n b m = a nb m a m b n b n b m a nb m a m b m + a m b m a m b n b n b m a nb m a m b m + a m b m a m b n κ 2 M κ 2 ( a n a m + b m b n ) 7

ε > 0 N 1, N 2 n, m N 1 = a n a m < κ2 ε 2M, n, m N 2 = b n b m < κ2 ε 2M N = max(n 1, N 2 ) n, m N = a n a m < κ2 ε 2M, b n b m < κ2 ε 2M n, m N a n a m b n M κ 2 ( a n a m + b m b n ) ε {a n /b n } n=1 1.11. {a n } n=1, {b n} n=1 b m 1. {a n } n=1 + {b n} n=1 {a n + b n } n=1 2. {a n } n=1 {b n} n=1 {a n b n } n=1 3. {a n } n=1 {b n} n=1 {a nb n } n=1 4. {b n } n=1 0 {a n} n=1 {b n} n=1 {a n/b n } n=1 1.12. A B A := B, A B 1.13. {a n } n=1, {b n} n=1, {a n} n=1, {b n} n=1 {a n} n=1 {a n} n=1 {b n } n=1 {b n} n=1 1. {a n } n=1 + {b n} n=1 {a n} n=1 + {b n} n=1 2. {a n } n=1 {b n} n=1 {a n} n=1 {b n} n=1 3. {a n } n=1 {b n} n=1 {a n} n=1 {b n} n=1 4. {b n } n=1 {b n} n=1 0 {a n} n=1 {b n } n=1 {a n} n=1 {b n} n=1 4 4 4. M a n + a n + b n + b n M {b n } n=1 {b n} n=1 0 κ > 0 b n, b n κ 8

ε > 0 N 1, N 2 n N 1 = a n a n < κ2 2M ε n N 2 = b n b n < κ2 2M ε a n a n b n = a nb n a nb n b n b n b n a nb n a nb n + a nb n a nb n b n b n a nb n a nb n + a nb n a nb n κ 2 M κ 2 ( a n a n + b n b n ) n N = max(n 1, N 2 ) a n a n b n M κ 2 ( a n a n + b n b n ) < ε b n {a n } n=1 {b n} n=1 {a n} n=1 {b n} n=1 2 2 2.1. {a n } n=1 {b n} n=1 {a n} n=1 {b n} n=1 ε > 0 N n N b n a n > ε 2.2. {a n } n=1, {b n} n=1, {a n} n=1, {b n} n=1 {a n} n=1 {a n} n=1 {b n } n=1 {b n} n=1 {a n} n=1 {b n} n=1 {a n} n=1 {b n} n=1. {a n } n=1 {b n} n=1 ε > 0 N n N b n a n > 1 3 ε {a n} n=1 {a n} n=1 {b n } n=1 {b n} n=1 N 1, N 2 n N 1 = a n a n < 1 3 ε n N 2 = b n b n < 1 3 ε 9

n N = max(n 1, N 2, N) b n a n = b n b n + b n a n + a n a n > 1 3 ε 1 3 ε 1 3 ε = ε {a n} n=1 {b n} n=1 2.3. {a n } n=1, {b n} n=1, {c n} n=1 1. {a n } n=1 {a n} n=1 2. {a n } n=1 {b n} n=1 {b n} n=1 {a n} n=1 {a n} n=1 = {b n} n=1 3. {a n } n=1 {b n} n=1 {b n} n=1 {c n} n=1 {a n} n=1 {c n} n=1 4. {a n } n=1 {b n} n=1 {b n} n=1 {a n} n=1 5. {a n } n=1 {b n} n=1 {a n} n=1 + {c n} n=1 {b n} n=1 + {c n} n=1 6. 0 {a n } n=1 {b n} n=1 {c n} n=1 {a nb n } n=1 {a nc n } n=1. 4 {a n } n=1 {b n} n=1 ε 0 > 0 N n 0 N b n0 a n0 ε 0 {a n } n=1 {b n} n=1 ε > 0 N 1, N 2 n, m N 1 = a n a m < 1 3 ε 0 n, m N 2 = b n b m < 1 3 ε 0 N = max(n 1, N 2 ) n 0 N a n0 b n0 ε 0 m N b m a m = (b m b n0 ) + (b n0 a n0 ) + (a n0 a m ) < 1 3 ε 0 ε 0 + 1 3 ε 0 = 1 3 ε 0 ε 0 N m N = b m a m < 1 3 ε 0 < 0 ε > 0 n N = a n b n > 1 3 ε 0 > ε {a n } n=1 {b n} n=1 10

3 3.1 ( ). {{a m,n } n=1 } m=1 m N {a m,n } n=1 3.2 ( ). {α n } m=1 = {{a m,n } n=1 } m=1 ε = {ε n} n=1 0 N n n, m > N ε < α m α n < ε 3.3 ( ). α = {α n } n=1 {α n } m=1 = {{a m,n} n=1 } m=1 α ε = {ε n } n=1 0 N n n, m > N ε < α m α < ε α {α n } m=1 = {{a m,n} n=1 } m=1 {α n } m=1 = {{a m,n} n=1 } m=1 3.4 ( ). α = {a n } n=1 α m = {a m } n=1 α Proof. ε = {ε n } n=1 0 ε 0 N 0 n N 0 ε n ε 0 > 0 ε > 0 {a n } n=1 N m N 0 lim m α m = α n, m N = a n a m < min(ε, ε 0 ) α α m < min(ε, ε 0 ) < ε 3.5.. {α m } m=1 = {{a m,n} n=1 } m=1 α m = {a m,n } n=1 N(m) n 1, n 2 N(m) 11

a m,n1 a m,n2 < 2 m {α m } m=1 L N M(L) m 1, m 2 M(L) 2 L 2 < α m1 α m2 < 2 L 2 m 1, m 2 M(L) Ñ(m 1, m 2 ) m 1, m 2 M(L), n > Ñ(m 1, m 2 ) 2 L 1 < α m1,n α m2,n < 2 L 2 2 L 1 < α m1,n(m 1 )+N(m 2 )+Ñ(m 1,m 2 ) α m 2,N(m 1 )+N(m 2 )+Ñ(m 1,m 2 ) < 2 L 1 2 m 1 < α m1,n(m 1 ) α m1,n(m 1 )+N(m 2 )+Ñ(m 1,m 2 ) < 2 m 1 2 m 2 < α m2,n(m 2 ) α m2,n(m 1 )+N(m 2 )+Ñ(m 1,m 2 ) < 2 m 2 m 1, m 2 L + 2 2 L < α m1,n(m 1 ) α m2,n(m 2 ) < 2 L ε > 0 2 L < ε L L m 1, m 2 L + 2 ε < α m1,n(m 1 ) α m2,n(m 2 ) < ε α = {a L,N(L) } L=1 lim m α m = α ε > 0 L N 2 L < ε α m α = {α m,n α n,n(n) } n=1 m n N(m) α m,n α n,n(n) = α m,n α m,n(m) + α m,n(m) α n,n(n) m, n L + 2 2 L < α m,n(m) α n,n(n) < 2 L N(m) n N(m) 2 m α m,n α m,n(m) < 2 m m L + 2 n L + 2 2 L+1 < α m,n α n,n(n) < 2 L+1 m L + 2 2 L+1 < α m α < 2 L+1 lim α m = α m 12

3.6. K = {k n } n=1 {α m} = {{a m,n } n=1 } α 1 α 2 α m α m+1 K α = {α m } m=1. α ε > 0 M m 1, m 2 m 1 > m 2 M α m1 α m2 > ε {α m } m=1 ε α m m K 3.7. A M a A a M M M 0 M 0 1. a A a M 0 2. m < M 0 a A a > m. {M n } n=1 M 1 M 0 n, n = 0, 1, 2, M 1, M 2,, M n M n+1 1. a A a M n 2 n M n+1 = M n 2 n 2. M n+1 = M n M n 2 n M n+1 M n {M n } n=1 M 1 1 M n M = lim n M n a M a M n, n = 1, 2,, a A 13

4 Q (a) R i. 0 0 + α = α + 0. ii. α α α+( α) = ( α)+α = 0. iii. α, β, γ α + (β + γ) = (α + β) + γ. iv. α, β α + β = β + α (b) i. 1 α1 = 1α ii. α, β, γ α(βγ) = (αβ)γ. iii. α, β, γ α(β+γ) = αβ+αγ (α+β)γ = αγ + βγ. iv. α, β αβ = βα. (c) i. 0 α αα 1 = α 1 α = 0. ii. 1 0. (a) i. α α. ii. α β, β α α = β. iii. α β, β γ α γ. iv. α β β α (b) α β α + γ β + γ. (c) α β, 0 γ αγ βγ. 1. R 2. Q Q Q π 14