Definitions

Question 1

Bounded (Sequences)

Answer 1

A sequence (xn) s bounded if there is an M ∈ R such that |xn|<M for all n ∈ N

Question 2

Bounded (Sets)

Answer 2

A set S of real number is bounded above if there is a real number k such that k≥s for all s ∈ S.
The number k is called the upper bound of S. The terms bounded from below and lower bounds are similarly defined. A set is bounded if it has both upper and lower bounds. Therefore, a set of real number is bounded if it is contained in a finite interval.

Question 3

Bounded Above and Upper Bound

Answer 3

A set AcR is bounded above if there exists a number b ∈ R such that a≤b for all a ∈ A. The number b is called an upper bound for A.

Question 4

Bounded Below and Lower Bound

Answer 4

A set A is bounded below if there exists a lower bound l ∈ R satisfying l≤a for every a ∈ R.

Question 5

Maximum

Answer 5

A real number a0 is a maximum of the set A if a0 is an element of A and a0≥a for all a ∈ A.

Question 6

Minimum

Answer 6

A real number a1 is a minimum of A if a1 ∈ A and a1≤a for all a ∈ A.

Question 7

Supremum

Answer 7

A real number s is the supremum (or least upper bound) of a Set AcR if:

1. s is an upper bound of A
2. If b is any upper bound of A, then s≤b

Question 8

Infimum

Answer 8

A real number i is the greatest lower bound, or infimum for a set AcR if:

1. i is a lower bound for A: i≤a for all a ∈ A
2. If l is any lower bound for A, then l≤i

Question 9

Axiom of Completeness

Answer 9

Every non-empty set of real numbers that is bounded above has a least upper bound

Question 10

Converges to L

Answer 10

For any ε>0 there is a N ∈ N such that |an-b|<ε whenever n≥N.
If a sequence (an) converges to L then every subsequence converges to L as well.

Question 11

Converges (Sequences)

Answer 11

A sequence (an) converges to a real number a if, for every positive number ε, there exiss an N ∈ N such that when n≥N, it follows that |an-a|< ε.

Question 12

Convergence (of a Sequence) Topological version

Answer 12

A sequence convergences to a if, given any ε-neighborhood Vε(a) of a, there exists a point in the sequence after which all the terms are in Vε(a). In other words, every ε-neighborhood contains all but a finite number of terms of (an).

Question 13

Convergence (Series)

Answer 13

Let bn be a series of real numbers. An infinite series is a fomral expression of the form ∑bn = b1+b2+…
We definte a corresponding series of partial sums sm = b1+b2+…+bm = ∑bn (for n=1 to m)

If ∑ak = A then lim sm = A as well.

Question 14

Open and Closed Sets

Answer 14

Let S c R.

1) S is said to be open if every point of S is an interior point of S.
2) S is said to be closed iff R\S is open.

Question 15

Cauchy

Answer 15

A sequence (an) is Cuachy if for any ε>0 there is a N ∈ N such that   |an-am|<ε for all n, m ≥ N.
For any distance ε there isa point in the sequence past which all subsequent terms are within ε of each other.

Question 16

Monotone

Answer 16

A sequence (an) is increasing if
an ≤ an+1 for all n∈N and decreasing if
an ≥ an+1 for all n∈N. A sequence is monotone if it is either increasing or decreasing.

Question 17

Nested Intervals

Answer 17

For each n ∈ N, assume we aregiven a closed interval In = [an, bn]. Assume also that each In contains In+1 such that
I1≥I2≥I3≥…≥In≥…
Then I1 and I2 and I3 and … and In and …. ≠ the empty set

Question 18

Countable

Answer 18

A set A is countable if N~A. In other words, a set A is countable if there exists a bijection (one to one and onto) from the natural numbers to the set A.