Chapter 3

Question 1

an open ball of radius r centered at a, in a metric space

Answer 1

Let X be a metric space and let a be an element of x for r in R, define B_r(a): {x in X: d(a,x)<r></r>

<p>All x in X are within r of a</p>

</r>

Question 2

an open set, in a metric space

Answer 2

a subset u of a metric space X is _______ if u is the union of open balls

Question 3

diameter of a set S, in a metric space

Answer 3

Let X be a metric space and let S b contained in X.

1. If S =ø then diam(S)=0
2. Otherwise diam(S)= sup{d(x,y): x,y are in S} IF sup(S) exists

Question 4

the convergence of a sequence (an) to a limit x, in a metric space

Answer 4

The sequence (a_n) in a metric space (X,d) _______ to ________ ________ x in X if

for all N in the Naturals st. n>N, a_n is in B_epsilon(x)

d(x,a_n)<epsilon></epsilon>

<p>A tail of the sequence converges to x.</p>

</epsilon>

Question 5

the Nth tail of a sequence (a_n), in a metric space

Answer 5

Let (a_n) be a sequence in (X,d). Let N be an element of the Natural Numbers. The ____ _______ ____ __ ___________ is the subsequence (a_n)_{n greater than or equal to N}

a_N, a_N+1, a_N+2,…

Question 6

Theorem 3.4.10

Answer 6

Let S be a subset of R (S is contained in R) and supS=L. There exists an S_n in S s.t. S_n→L

Question 7

Theorem 3.4.11 Let a_n and b_n be a sequences in R. Let a_n→a and b_n→b. Then, (four things)

Answer 7

1. (a_n + or - b_n)→ a + or - b
2. (a_n•b_n)→ ab
3. (ka_n)→ ka
4. (a_n÷b_n)→ a÷b where b does not equal 0

Question 8

Let X be a metric space and let S be contained in X

There exists a sequence S_n contained in S whose elements do not include {x} that converges to x

Answer 8

limit point of a set S in a metric space X

Question 9

A closed set in a metric space

Answer 9

A set C contained in X is _______ if C contains its limit points

Question 10

Theorem 3.6.3 Let X be a metric space. Let C eb contained in X TFAE:

Can be used to prove that a closed ball is closed

Answer 10

TFAE:

1. C is closed
2. If (a_n) is a sequence in C converging to x, then x is an element of C. For all a_n→x, then x is an element of c

Question 11

Closed ball of radius r centered at a in a metric space X

Answer 11

Let X be a metric space and let a be an element of X for r in R we define C_r(a)= {x in X: d(a,x) is less than or equal to r}

Question 12

Boundedness in a metric space

Answer 12

S is bounded in a metric space X if diam(S) exists

Question 13

Theorem 3.1.12 about S bounded in X TFAE

Answer 13

Let S be contained in X. TFAE:

1. diamS exists (S is bounded)
2. There exists an a in X and there exists an r>0 B_r(a) s.t. S is contained in B_r(a)
3. For all elements X there exists a B_r(a) r>0 s.t S is contained in B_r(a)
   
   1. for all s in S, d(a,s)<r></r>
   </r>

Question 14

Hausdorff Property

Answer 14

for any x, y ∈ X with x =/= y there exist open sets U containing x and V containing y such that U intersect V = ∅.

the intersection of the distance between x and y is empty. The intersection of two disjoint open sets is empty

Question 15

discrete

Answer 15

If every {x} is open for all x in X, then X is ________

Every point is isolated

Question 16

Isolated

Answer 16

X is a metric space. x is in X. If {x} is open, then x is isolated. You must be able to fit an open interval around the point

Question 17

Theorem 3.4.3

Answer 17

Let a_n be a sequence in R converging to x. Then |a_n|→|x|.

1. There exists a k in R s.t a_n is greater than or equal to k, then x is greater than or equal to k and |a_n|→|x|
2. If a_n is less than or equal to k, then x is less than or equal to k

Question 18

Theorem 3.4.6/7

Answer 18

Let a_n and b_n be sequences in R s.t a_n→a and b_n→b

1. If a_n is less than or equal to b_n for all n, then a is less than b
2. If a_n is greater than or equal to b_n for all n, then a is greater than or equal to b

Question 19

Squeeze Theorem

Answer 19

Let a_n→x and b_n→x in R. Sps there exists c_n s.t. a_n is less than or equal to c_n is less than or equal to b_n, then c_n→x

Question 20

Monotone Sequence Theorem

Answer 20

If a_n is in R and

1. a_n is bounded
2. a_n is monotone

a_n is convergent

Every monotone sequence is convergent

Question 21

Every convergent sequence in R has a _______ _______

Answer 21

bounded subsequence

Question 22

There is a sequence of distinct points (non-repeating values) in S that converges to x

Answer 22

limit point of a set S in a metric space X

Question 23

Let X be a metric space and S be contained in X.

For all r>0, B_r(x) contains infinitely many points of S

Answer 23

limit point of a set S in a metric space X

Question 24

Let X be a metric space and S be contained in X.

For all r>0, B_r(x) contains one point in S{x}

Answer 24

limit point of a set S in a metric space X

Question 25

A set is closed if

Answer 25

it contains its limit points