Advanced Calculus (II)

Advanced Calculus (II)

W^EN-C^HINGL^IEN

Department of Mathematics National Cheng Kung University

2009

Ch9: Convergence in R

9.1: Limits Of Sequences

Definition (9.1)

Let {xk} be a sequence points in Rⁿ.

(i) {xk} is said to converge to some point a ∈ Rⁿ(call the limit ofx_k)if and only if for every ε > 0 there is an N ∈N such that

k ≥ N implies kx_k − ak < ε.

(ii) {xk} is said to be bounded if and only if there is an M > 0 such that kxkk ≤ M for all k ∈ N.

(iii)xk ∈ Rⁿ is said to be Cauchy if and only if for every ε >0 there is an N ∈N such that

k , m ≥ N imply kx_k − x_mk < ε.

WEN-CHINGLIEN Advanced Calculus (II)

Leta := (a(1), . . . , a(n)) and xk := (xk(1), . . . , xk(n)) belong toRⁿ for k ∈N. Then xk → a, as k → ∞, if and only if the component sequences xk(j) → a(j), as k → ∞, for all j = 1, 2, . . . , n.

Proof.

Fix j ∈ {1, . . . , n}. By Remark 8.7, kx_k(j) − a(j)k ≤ kx_k − ak ≤√

n max

1≤`≤n|x<sub>k</sub>(`) −a(`)|.

Hence, by the Squeeze Theorem, xk(j) → a(j) as k → ∞ for all 1 ≤ j ≤ n if and only if the real sequence

kx_k − ak → 0 as k → ∞. Since kx_k − ak → 0 if and only if xk → a, as k → ∞, the proof of the theorem is complete.

Theorem (9.2)

Leta := (a(1), . . . , a(n)) and xk := (xk(1), . . . , xk(n)) belong toRⁿ for k ∈N. Then xk → a, as k → ∞, if and only if the component sequences xk(j) → a(j), as k → ∞, for all j = 1, 2, . . . , n.

Proof.

Fix j ∈ {1, . . . , n}. By Remark 8.7, kx_k(j) − a(j)k ≤ kxk − ak ≤√

n max

1≤`≤n|x<sub>k</sub>(`) −a(`)|.

Hence, by the Squeeze Theorem, xk(j) → a(j) as k → ∞ for all 1 ≤ j ≤ n if and only if the real sequence

kx_k − ak → 0 as k → ∞. Since kx_k − ak → 0 if and only if xk → a, as k → ∞, the proof of the theorem is complete.

WEN-CHINGLIEN Advanced Calculus (II)

Leta := (a(1), . . . , a(n)) and xk := (xk(1), . . . , xk(n)) belong toRⁿ for k ∈N. Then xk → a, as k → ∞, if and only if the component sequences xk(j) → a(j), as k → ∞, for all j = 1, 2, . . . , n.

Proof.

Fix j ∈ {1, . . . , n}. By Remark 8.7, kx_k(j) − a(j)k ≤ kx_k − ak ≤√

n max

1≤`≤n|x<sub>k</sub>(`) −a(`)|.

Hence, by the Squeeze Theorem, xk(j) → a(j) as k → ∞ for all 1 ≤ j ≤ n if and only if the real sequence

kx_k − ak → 0 as k → ∞. Since kx_k − ak → 0 if and only if xk → a, as k → ∞, the proof of the theorem is complete.

Theorem (9.2)

Leta := (a(1), . . . , a(n)) and xk := (xk(1), . . . , xk(n)) belong toRⁿ for k ∈N. Then xk → a, as k → ∞, if and only if the component sequences xk(j) → a(j), as k → ∞, for all j = 1, 2, . . . , n.

Proof.

Fix j ∈ {1, . . . , n}. By Remark 8.7, kx_k(j) − a(j)k ≤ kxk − ak ≤√

n max

1≤`≤n|x<sub>k</sub>(`) −a(`)|.

Hence, by the Squeeze Theorem,xk(j) → a(j) as k → ∞ for all 1 ≤ j ≤ n if and only if the real sequence

kx_k − ak → 0 as k → ∞. Since kx_k − ak → 0 if and only if xk → a, as k → ∞, the proof of the theorem is complete.

WEN-CHINGLIEN Advanced Calculus (II)

Leta := (a(1), . . . , a(n)) and xk := (xk(1), . . . , xk(n)) belong toRⁿ for k ∈N. Then xk → a, as k → ∞, if and only if the component sequences xk(j) → a(j), as k → ∞, for all j = 1, 2, . . . , n.

Proof.

Fix j ∈ {1, . . . , n}. By Remark 8.7, kx_k(j) − a(j)k ≤ kxk − ak ≤√

n max

1≤`≤n|x<sub>k</sub>(`) −a(`)|.

Hence, by the Squeeze Theorem, xk(j) → a(j) as k → ∞ for all 1 ≤ j ≤ n if and only if the real sequence

kx_k − ak → 0 as k → ∞. Since kx_k − ak → 0 if and only if xk → a, as k → ∞, the proof of the theorem is complete.

Theorem (9.2)

Leta := (a(1), . . . , a(n)) and xk := (xk(1), . . . , xk(n)) belong toRⁿ for k ∈N. Then xk → a, as k → ∞, if and only if the component sequences xk(j) → a(j), as k → ∞, for all j = 1, 2, . . . , n.

Proof.

Fix j ∈ {1, . . . , n}. By Remark 8.7, kx_k(j) − a(j)k ≤ kxk − ak ≤√

n max

1≤`≤n|x<sub>k</sub>(`) −a(`)|.

Hence, by the Squeeze Theorem,xk(j) → a(j) as k → ∞ for all 1 ≤ j ≤ n if and only if the real sequence

kx_k − ak → 0 as k → ∞. Since kx_k − ak → 0 if and only if xk → a, as k → ∞, the proof of the theorem is complete.

WEN-CHINGLIEN Advanced Calculus (II)

Leta := (a(1), . . . , a(n)) and xk := (xk(1), . . . , xk(n)) belong toRⁿ for k ∈N. Then xk → a, as k → ∞, if and only if the component sequences xk(j) → a(j), as k → ∞, for all j = 1, 2, . . . , n.

Proof.

Fix j ∈ {1, . . . , n}. By Remark 8.7, kx_k(j) − a(j)k ≤ kxk − ak ≤√

n max

1≤`≤n|x<sub>k</sub>(`) −a(`)|.

Hence, by the Squeeze Theorem, xk(j) → a(j) as k → ∞ for all 1 ≤ j ≤ n if and only if the real sequence

kx_k − ak → 0 as k → ∞. Since kx_k − ak → 0 if and only if xk → a, as k → ∞, the proof of the theorem is complete.

Theorem (9.2)

Leta := (a(1), . . . , a(n)) and xk := (xk(1), . . . , xk(n)) belong toRⁿ for k ∈N. Then xk → a, as k → ∞, if and only if the component sequences xk(j) → a(j), as k → ∞, for all j = 1, 2, . . . , n.

Proof.

Fix j ∈ {1, . . . , n}. By Remark 8.7, kx_k(j) − a(j)k ≤ kxk − ak ≤√

n max

1≤`≤n|x<sub>k</sub>(`) −a(`)|.

Hence, by the Squeeze Theorem, xk(j) → a(j) as k → ∞ for all 1 ≤ j ≤ n if and only if the real sequence

kx_k − ak → 0 as k → ∞. Since kx_k − ak → 0 if and only if xk → a, as k → ∞, the proof of the theorem is complete.

WEN-CHINGLIEN Advanced Calculus (II)

For eacha ∈ Rⁿthere is a sequence xk ∈ Qⁿ such that xk → a as k → ∞.

Theorem (9.4) Let n ∈N.

(i) A sequence inRⁿ can have at most one limit.

(ii) If {x_k}k ∈Nis a sequence inRⁿthat converges toa and {xk_j}k ∈N is any subsequence of {xk}k ∈N, thenxk_j

converges toa as j → ∞.

(iii) Every convergent sequence inRⁿis bounded, but not conversely.

(iv) Every convergent sequence inRⁿis Cauchy.

WEN-CHINGLIEN Advanced Calculus (II)

Let n ∈N.

(i) A sequence inRⁿ can have at most one limit.

(ii) If {x_k}k ∈Nis a sequence inRⁿthat converges toa and {xk_j}k ∈N is any subsequence of {xk}k ∈N, thenxk_j

converges toa as j → ∞.

(iii) Every convergent sequence inRⁿis bounded, but not conversely.

(iv) Every convergent sequence inRⁿis Cauchy.

Theorem (9.4) Let n ∈N.

(i) A sequence inRⁿ can have at most one limit.

(ii) If {x_k}k ∈Nis a sequence inRⁿthat converges toa and {xk_j}k ∈N is any subsequence of {xk}k ∈N, thenxk_j

converges toa as j → ∞.

(iii) Every convergent sequence inRⁿis bounded, but not conversely.

(iv) Every convergent sequence inRⁿis Cauchy.

WEN-CHINGLIEN Advanced Calculus (II)

Let n ∈N.

(i) A sequence inRⁿ can have at most one limit.

(ii) If {x_k}k ∈Nis a sequence inRⁿthat converges toa and {xk_j}k ∈N is any subsequence of {xk}k ∈N, thenxk_j

converges toa as j → ∞.

(iii) Every convergent sequence inRⁿis bounded, but not conversely.

(iv) Every convergent sequence inRⁿis Cauchy.

Theorem (9.4)

(v) If {xk} and {y_k} are convergent sequences in Rⁿ and α ∈R, then

k →∞lim(xk +yk) = lim

k →∞xk+ lim

k →∞yk,

k →∞lim(αxk) = α lim

k →∞(xk), and

k →∞lim(xk · y_k) =



k →∞lim(xk)



·



k →∞lim(yk)

 . Moreover, when n = 3,

k →∞lim(xk × yk) =



k →∞lim(xk)



×



k →∞lim(yk)

 .

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (9.5 Bolzano-Weierstrass Theorem forR ) Every bounded sequence inRⁿhas a convergent subsequence.

Theorem (9.6)

A sequence {xk} in Rⁿis Cauchy if and only if it converges.

WEN-CHINGLIEN Advanced Calculus (II)

Letxk ∈ Rⁿ.Thenxk → a as k → ∞ if and only if for every open set V that containsa there is an N ∈ N such that k ≥ N impliesxk ∈ V .

Theorem (9.8)

Let E ⊆Rⁿ.Then E is closed if and only if E contains all its limit points; i.e.,xk ∈ E and xk → x imply x ∈ E.

WEN-CHINGLIEN Advanced Calculus (II)

Let E be a subset ofRⁿ.

(i) An open covering of E is a collection of sets {Vα}α∈A

such that each Vα is open and E ⊆ [

α∈A

Vα.

(ii) The set E is said to be compact if and only if every open covering of E has a finite subcovering; i.e., if {V_α}_α∈A is an open covering of E , then there is a finite subset A0 of A such that

E ⊆ [

α∈A₀

Vα.

Lemma (9.10 Borel Covering Lemma)

Let E be a closed, bounded subset ofRⁿ.If r is any function from E into (0, ∞), then there exist finitely many pointsy1, . . . ,yN ∈ E such that

E ⊆

N

[

j=1

Br (y_j)(yj).

WEN-CHINGLIEN Advanced Calculus (II)

Let E be a subset ofRⁿ.Then E is compact if and only if E is closed and bounded.

Thank you.

WEN-CHINGLIEN Advanced Calculus (II)