## Advanced Calculus (II)

W^{EN}-C^{HING}L^{IEN}

Department of Mathematics National Cheng Kung University

2009

## Ch9: Convergence in **R**

^{n}

### 9.1: Limits Of Sequences

Definition (9.1)

Let {xk**} be a sequence points in R**^{n}.

(i) {xk**} is said to converge to some point a ∈ R**^{n}(call the
limit of**x**_{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 kxk**k ≤ M for all k ∈ N.**

(iii)**x**k **∈ R**^{n} 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**_{m}k < ε.

WEN-CHINGLIEN **Advanced Calculus (II)**

Let**a := (a(1), . . . , a(n)) and x**k := (xk(1), . . . , xk(n))
belong to**R**^{n} for k ∈**N. Then x**k **→ 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_{k}(`) −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**
**x**k **→ a, as k → ∞, the proof of the theorem is complete.**

Theorem (9.2)

Let**a := (a(1), . . . , a(n)) and x**k := (xk(1), . . . , xk(n))
belong to**R**^{n} for k ∈**N. Then x**k **→ 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_{k}(`) −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**
**x**k **→ a, as k → ∞, the proof of the theorem is complete.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Let**a := (a(1), . . . , a(n)) and x**k := (xk(1), . . . , xk(n))
belong to**R**^{n} for k ∈**N. Then x**k **→ 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_{k}(`) −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**
**x**k **→ a, as k → ∞, the proof of the theorem is complete.**

Theorem (9.2)

**a := (a(1), . . . , a(n)) and x**k := (xk(1), . . . , xk(n))
belong to**R**^{n} for k ∈**N. Then x**k **→ 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_{k}(`) −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**
**x**k **→ a, as k → ∞, the proof of the theorem is complete.**

WEN-CHINGLIEN **Advanced Calculus (II)**

**a := (a(1), . . . , a(n)) and x**k := (xk(1), . . . , xk(n))
belong to**R**^{n} for k ∈**N. Then x**k **→ 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_{k}(`) −a(`)|.

**kx**_{k} **− ak → 0 as k → ∞.** Since kx_{k} **− ak → 0 if and only if**
**x**k **→ a, as k → ∞, the proof of the theorem is complete.**

Theorem (9.2)

**a := (a(1), . . . , a(n)) and x**k := (xk(1), . . . , xk(n))
belong to**R**^{n} for k ∈**N. Then x**k **→ 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_{k}(`) −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**
**x**k **→ a, as k → ∞, the proof of the theorem is complete.**

WEN-CHINGLIEN **Advanced Calculus (II)**

**a := (a(1), . . . , a(n)) and x**k := (xk(1), . . . , xk(n))
belong to**R**^{n} for k ∈**N. Then x**k **→ 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_{k}(`) −a(`)|.

**kx**_{k} **− ak → 0 as k → ∞.** Since kx_{k} **− ak → 0 if and only if**
**x**k **→ a, as k → ∞, the proof of the theorem is complete.**

Theorem (9.2)

**a := (a(1), . . . , a(n)) and x**k := (xk(1), . . . , xk(n))
belong to**R**^{n} for k ∈**N. Then x**k **→ 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_{k}(`) −a(`)|.

**kx**_{k} **− ak → 0 as k → ∞. Since kx**_{k} **− ak → 0 if and only if**
**x**k **→ a, as k → ∞, the proof of the theorem is complete.**

WEN-CHINGLIEN **Advanced Calculus (II)**

For each**a ∈ R**^{n}there is a sequence **x**k **∈ Q**^{n} such that
**x**k **→ a as k → ∞.**

Theorem (9.4)
Let n ∈**N.**

(i) A sequence in**R**^{n} can have at most one limit.

(ii) If {x_{k}}k ∈Nis a sequence in**R**^{n}that converges to**a and**
**{x**k_{j}}k ∈N is any subsequence of {xk}k ∈N, then**x**k_{j}

converges to**a as j → ∞.**

(iii) Every convergent sequence in**R**^{n}is bounded, but not
conversely.

(iv) Every convergent sequence in**R**^{n}is Cauchy.

WEN-CHINGLIEN **Advanced Calculus (II)**

Let n ∈**N.**

(i) A sequence in**R**^{n} can have at most one limit.

(ii) If {x_{k}}k ∈Nis a sequence in**R**^{n}that converges to**a and**
**{x**k_{j}}k ∈N is any subsequence of {xk}k ∈N, then**x**k_{j}

converges to**a as j → ∞.**

(iii) Every convergent sequence in**R**^{n}is bounded, but not
conversely.

(iv) Every convergent sequence in**R**^{n}is Cauchy.

Theorem (9.4)
Let n ∈**N.**

(i) A sequence in**R**^{n} can have at most one limit.

(ii) If {x_{k}}k ∈Nis a sequence in**R**^{n}that converges to**a and**
**{x**k_{j}}k ∈N is any subsequence of {xk}k ∈N, then**x**k_{j}

converges to**a as j → ∞.**

(iii) Every convergent sequence in**R**^{n}is bounded, but not
conversely.

(iv) Every convergent sequence in**R**^{n}is Cauchy.

WEN-CHINGLIEN **Advanced Calculus (II)**

Let n ∈**N.**

(i) A sequence in**R**^{n} can have at most one limit.

_{k}}k ∈Nis a sequence in**R**^{n}that converges to**a and**
**{x**k_{j}}k ∈N is any subsequence of {xk}k ∈N, then**x**k_{j}

converges to**a as j → ∞.**

(iii) Every convergent sequence in**R**^{n}is bounded, but not
conversely.

(iv) Every convergent sequence in**R**^{n}is Cauchy.

Theorem (9.4)

(v) If {xk**} and {y**_{k}**} are convergent sequences in R**^{n} and
α ∈**R, then**

k →∞lim(xk +**y**k) = lim

k →∞**x**k+ lim

k →∞**y**k,

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 **× y**k) =

k →∞lim(xk)

×

k →∞lim(yk)

.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (9.5 Bolzano-Weierstrass Theorem for**R** )
Every bounded sequence in**R**^{n}has a convergent
subsequence.

Theorem (9.6)

A sequence {xk**} in R**^{n}is Cauchy if and only if it
converges.

WEN-CHINGLIEN **Advanced Calculus (II)**

Let**x**k **∈ R**^{n}.Then**x**k **→ a as k → ∞ if and only if for**
every open set V that contains**a there is an N ∈ N such**
that k ≥ N implies**x**k ∈ V .

Theorem (9.8)

Let E ⊆**R**^{n}.Then E is closed if and only if E contains all
its limit points; i.e.,**x**k **∈ E and x**k **→ x imply x ∈ E.**

WEN-CHINGLIEN **Advanced Calculus (II)**

Let E be a subset of**R**^{n}.

(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∈A_{0}

Vα.

Lemma (9.10 Borel Covering Lemma)

Let E be a closed, bounded subset of**R**^{n}.If r is any
function from E into (0, ∞), then there exist finitely many
points**y**1, . . . ,**y**N ∈ E such that

E ⊆

N

[

j=1

Br (y_{j})(yj).

WEN-CHINGLIEN **Advanced Calculus (II)**

Let E be a subset of**R**^{n}.Then E is compact if and only if
E is closed and bounded.

## Thank you.

WEN-CHINGLIEN **Advanced Calculus (II)**