Test 2

Question 1

Definition of a subsequence

Answer 1

If {Nk} is a strictly increasing sequence of positive integers, then the sequence {Xnk} is a subsequence of {Xn}

Question 2

Thm of convergent subsequences

Answer 2

Suppose {Xn} ⊆ R is a sequence with Xn-> a∈R as n -> ∞. Then every subsequence of {Xn} convergest to a.

Question 3

Bolzano-Weierstrass Theorem

Answer 3

Every bounded sequence in R has a convergent subsequence.

Question 4

Definition of Cauchy Sequence

Answer 4

A sequence {Xn} is a Cauchy sequence if for all ε>0 there exists an N∈N such that for all m,n >= N |Xn-Xm|<ε

Question 5

Cauchy Criterion Theorem

Answer 5

A sequence {Xn}⊆R is cauchy if and only if {Xn} converges

Question 6

Definition of Cluster Point

Answer 6

For a sequence {Xn} a∈R is a cluster point of {Xn} if there is a subsequence of {Xn} converging to a.

Question 7

Definition of divergent sequence

Answer 7

A sequence {Xn} diverges to ∞ if for all M>0 there exists an N∈N such that for all n>=N, Xn>M.
The sequences diverges to -∞ if for all M<0 there exists an N∈N such that for all n>=N Xn<M

Question 8

Definition of Limit Superior

Answer 8

If {Xn}⊆R then the limit superior of {Xn} is defined as limsup Xn = lim(sup{Xk, k>=n} = inf(sup{Xk: k>=n})

Question 9

Definition of Limit Inferior

Answer 9

If {Xn}⊆R then the limit inferior of {Xn} is defined as liminf Xn = lim(inf{Xk, k>=n} = sup(inf{Xn : k>=n})

Question 10

Definition of an Open set

Answer 10

A set U⊆R is said to be open if for all x∈U, there exists an ε(x)>0 such that (x-ε, x+ε) ⊆ U

Question 11

Definition of Interior Point

Answer 11

Let E⊆R. A point x∈E is said to be an interior point of E if there exists an ε>0 such that (x-ε , x+ε )⊆E.

Question 12

Definition of an Index Set

Answer 12

An index set is a set whose members label (or index) members of another set.

Question 13

Defition of a Closed Set

Answer 13

A set F⊆R is said to be closed if F^c - R\F is open.

Question 14

Definition of a Limit Point

Answer 14

Let E⊆R. A point a∈R is called the limit point of E if for all ε>0, there exists a x∈E, x=/=a with |x-a|<ε

Question 15

Definition of Isolated Point

Answer 15

Let E⊆R. A point a∈E is called an Isolated Point if it is not a limit point.

Question 16

Theorem of closed sets

Answer 16

The set F⊆R is closed if and only if F contains all of its limit points

Question 17

Definition of Closure

Answer 17

For E⊆R, E’ denotes the set of limit points of E. The closure of E, Ē is Ē = E u E’

Question 18

Definition of a Dense set

Answer 18

A set D⊆R is Dense in R if D̄ = R

Question 19

Definition of a Connected set

Answer 19

A set A⊆R is connected if there does not exists two disjoint open sets U and V s.t. A ∩ U =/= {}, A ∩ V =/= {} and (A ∩ U) u (A ∩ V) = A

Question 20

Theorem of connected subsets

Answer 20

A subset A of R is connected if and only if it is an interval.

Question 21

Definition of an Open Cover

Answer 21

Let E⊆R. A collection U={U_alpha} of open subsets of R is called an open cover of E if E⊆∪{U_alpha}

Question 22

Definition of an Open Subcover

Answer 22

Let E⊆R and {U} be an open cover of E. If there exists B⊆A s.t. {Ub} is an open cover of E, {Ub} is an open subcover of {Ua}

Question 23

Definition of a compact subset

Answer 23

A subset K⊆R is compact if every open cover of K has a finite subcover.

Question 24

Theorem of compact subsets

Answer 24

A compact subset of R is closed and bounded

Question 25

Bolzano-Weierstrass Property

Answer 25

If S is an infinite subset of a compact set K, then S has a limit point in K

Question 26

Canters Intersection Theorem

Answer 26

If {Kn} is a sequence of nonempty compact subsets of R with Kn>=Kn+1 for all n∈N, then K=∩{Kn} is nonempty and compact

Question 27

Heine-Borel Theorem

Answer 27

Every closed and bounded interval [a,b] is compact

Question 28

Heine-Borel-Bolzano-Weierstrass Theorem

Answer 28

Let K⊆R. The following are equivalent. 1) K is closed and bounded 2) K is compact 3) Every infinite subset of K has a limit point in K

Question 29

Defintion of a Limit

Answer 29

Let E⊆R and f a real valued function with domain E. Let c be a limit point of E. Then f has a limit of L at c, denoted f(x)->L as x->c if for all ε>0, if 0<|x-c|<δ , then |f(x)-L|<ε

Question 30

Definition of δ-Neighborhood

Answer 30

Let δ>0, c∈R. (c-δ, c+δ) is called a δ-neighborhood of c.

Question 31

Definition of Deleted Neighborhood

Answer 31

Let δ>0, c∈R. (c-δ, c+δ) \ {c} is called a deleted neighborhood of c.

Question 32

Properties of Limits theorem

Answer 32

1) f(x)+g(x) -> L1+L2 2) f(x)g(x) -> L1L2 3) if L2=/=0, f(x)/g(x) -> L1/L2 as x->c

Question 33

Squeeze Theorem

Answer 33

Let Let E⊆R and f,g,h be real valued functions with domain E. Suppose f(x)<=g(x)<=h(x) on some deleted neighborhood of c. If f(x)->L and h(x)->L as x->c, then g(x)->L as x->c.