Exam 1

Question 1

r is rational if

Answer 1

r equals p over q such that p and q are integers and q is not equal to zero

Question 2

M is an upper bound if

Answer 2

For all x in the set S M is greater than or equal to x

Question 3

M is a lower bound if

Answer 3

For all x in the set S, M is less than or equal to x

Question 4

S is bounded above if

Answer 4

There exists a real number M such that for all x in the set S, M is greater than or equal to x

Question 5

M is bounded below if

Answer 5

There exists a real number M such that for all x in the set S M is greater than or equal to x

Question 6

M is the maximum of S if

Answer 6

1. M is in the set S

2. For all x in the set S, M is greater than or equal to x

Question 7

M is the supremum of S if

Answer 7

1. S is bounded above

2. For all upperbounds U, M is less than or equal to U

Question 8

M is the minimum of S if

Answer 8

1. M is in the set

2. For all x in the set S, M less than or equal to x

Question 9

M is the infimum of S if

Answer 9

1. S is bounded below

2. For all lower bounds B, M greater than or equal to B

Question 10

Xn converges to L if

Answer 10

For all epsilon greater than zero there exists a k in the natural number for all n greater than k such that the absolute value of xn- L is less than epsilon

Question 11

Xn diverges if

Answer 11

For all real numbers L, Xn does not converge to L

Question 12

Xn is bounded if

Answer 12

Exists M greater than zero for all n in the natural numbers such that the absolute of M is greater than Xn

Question 13

Xn is bounded below if

Answer 13

There exists a real number M for all n in the natural numbers such that M is less than Xn

Question 14

Xn is bounded above if

Answer 14

There exists a real number M for all n in the natural numbers such that Xn is less than M

Question 15

Xn is increasing if

Answer 15

For all natural numbers n Xn is less than or equal to X(n+1)

Question 16

Xn is decreasing if

Answer 16

For all n in the natural numbers, Xn is greater than or equal to X(n+1)

Question 17

Xn is a monotone sequence if

Answer 17

Xn is either decreasing or Xn is increasing

Question 18

Xn is a Cauchy sequence if

Answer 18

For all epsilon greater than zero there exists a k in the natural numbers for all n and m greater than k such that the absolute value Xn-Xm is less than epsilon

Question 19

L is a subsequential limit of Xn if

Answer 19

There is some subsequence of Xn converging to L

Question 20

Completeness Axiom

Answer 20

All non empty subset of the real numbers which are bounded above has a supremum

Ensures that the number line has no gaps

Question 21

Density Property of Q

Answer 21

For any two real numbers a and b such that a<b></b>

Question 22

The Limit Theorem

Answer 22

If Xn converges to A and Yn converges to B and r is a real number, then

1. rXn converges to rA
2. Xn + Yn converges to A+B
3. XnYn converges to AB
4. Xn/Yn converges to A/B

Question 23

Monotone Sequence Theorem

Answer 23

If Xn is decreasing and bounded below or increasing and bounded above, Xn converges

Question 24

Cauchy’s Theorem

Answer 24

Xn converges if and only if Xn is Cauchy

Question 25

Xn diverges to infinity if

Answer 25

For all M greater than zero there exists a k in the natural number for all n greater than k such that M less than Xn