Chapter 2

Question 1

sequence

Answer 1

a function x mapping the natural numbers to the reals. we usually denote the nth element of a sequence by x sub n

Question 2

converge

Answer 2

a sequence is said to converge to a number iif for every epsilon greater than zero there exists a M in the naturals such that |x sub n -x| < epsilon for all n greater than or equal to M. x is said to be the limit of the sequence.

Question 3

Facts about sequences

Answer 3

1. a convergent sequence has a unique limit
2. a convergent sequence is bounded
3. the sequence converges if and only if the k tail converges
4. if x sub n is a convergent sequence, then any subsequence is also convergent and the limits are the same

Question 4

monotone increasing

Answer 4

x sub n is less than or equal to x sub(n+1) for all n.

Question 5

monotone decreasing

Answer 5

x sub n is greater than or equal to all x sub (n+1)

Question 6

facts about monotone sequences

Answer 6

1. a monotone sequence is bounded if and only if it is convergent
2. the limit of a monotone increasing and bounded sequence is the supremum of the sequence
3. the limit of a monotone decreasing sequence is the infimum
4. let S in Reals be a nonempty bounded set. then there exist monotone sequences such that the limit of one is the supremum of S and the limit of the other is the infimum of the sequence

Question 7

K tail

Answer 7

tail of the sequence starting at K+1

Question 8

subsequence

Answer 8

Let {x sub n} be a sequence let {n sub i} be a strictly increasing sequence of natural numbers. The sequence {x sub n sub i} is called a subsequence of {x sub n}

Question 9

More facts about sequences

Answer 9

5. squeeze lemma: let {an}, {bn} and {xn} be sequences such that an is less than or equal to xn which is less than or equal to bn. Suppose an and bn converge to the same limit. Then xn converges and the limit of xn is equal to the limit of an and bn
6. let xn and yn be convergent sequences and xn less than or equal to yn for all n. Then, lim xn is less than or equal to lim yn.

Question 10

More More facts about sequences

Answer 10

7. let xn be a convergent sequence such that xn is greater than or equal to zero. Then lim xn is also greater than zero
8. let a, b in reals and let xn be a convergent sequence such that a is less than or equal to xn is less than or equal to b for all n. then the limit of xn is bounded by a and b

Question 11

Facts about Limits

Let xn and yn be convergent sequences

Answer 11

1. zn = yn+xn: lim(xn+yn)= lim xn +lim yn = lim zn
2. zn = xn-yn : lim (xn-yn) = lim xn -lim yn = lim zn
3. zn = xnyn : lim (xnyn) = lim(xn)lim(yn) = lim zn
4. if lim yn not equal to zero and yn not equal to zero ever, zn = xn/yn: lim (xn/yn) = limxn/limyn = lim zn

Question 12

Facts about convergent sequences

Answer 12

1. lim sqrt(xn) = sqrt(lim xn)
2. if {xn} convergent, {|xn|} convergent
3. lim |xn| = |lim xn|

Question 13

Convergence Tests

Answer 13

1. let {xn} be a sequence. suppose there is an x in the reals and a convergent sequence an such that lim an = 0 and |xn-x| less than or equal to an for all n. then xn converges and lim xn = x
2. let c>0. if c1, c^n is unbounded.
3. ratio test: L:=lim |x_{n+1}| /|xn| exists. if L1, xn unbounded

Question 14

limit superior

Answer 14

the limit of all the supremums of a sequence:
if {xn} is bounded, let an = sup{xk, k greater than or equal to n}. an is bounded monotone decreasing. lim sup xn = lim an

Question 15

limit inferior

Answer 15

let {xn} be a bounded sequence. bn = inf{xk” k greater than or equal to n}. bn is bounded monotone increasing. lim inf xn = lim bn

Question 16

Facts about limit inferior and limit superior
{xn} bounded sequence
{an} and {bn} are lim sup and lim inf, respectively

Answer 16

1. lim sup xn = inf {an: n in Naturals} an lim inf xn = sup{bn:n in Naturals}
2. lim inf xn less than or equal to lim sup xn
3. there exists a subsequence of {xn} such that the limit of the subsequence is lim sup xn
4. similarly, there exists a subsequence such that the limit of the subsequence = lim inf xn

Question 17

Using limit inferior and limit superior

Answer 17

1. let {xn} be a bounded sequence. then {xn} converges if and only if lim inf xn = lim sup xn
2. if {xn} converges, lim xn = lim inf xn = lim sup xn
3. if xn is a bounded sequence and xnk is a subsequence, then lim inf xn is less than or equal to lim inf xnk is less than or equal to lim sup xnk is less than or equal to lim sup xn
4. a bounded sequence is convergent and converges to x if and only if every convergent subsequence converges to x.

Question 18

Bolzano - Weierstrass Theorem

Answer 18

Suppose a sequence {xn} of real numbers is bounded. Then there exists a convergent subsequence {x_n_i}.

Question 19

Diverges to Infinity

Answer 19

we say xn diverges to infinity if for every m in reals there exists an N in the Naturals such that for all n greater than or equal to N, we have xn >m. if the case is that xn <m, we say xn diverges to minus infinity

Question 20

Alternate version of Bolzano Weierstrass

Answer 20

Let S in the reals be a bounded, infinite set. Then, there exists at least one cluster point of S.

Question 21

Cauchy Sequences

Answer 21

A sequence {xn} is a cauchy sequence if for every epsilon greater than zero, there exists an M in the naturals such that for all n greater than or equal to M and all k greater than or equal to M we have |xn-xk|<epsilon

Question 22

Facts about Cauchy Sequences

Answer 22

1. a cauchy sequence is bounded

2. a sequence of real numbers is cauchy if and only if it converges.