Chapter 2-4

Question 1

Closed

Answer 1

A set B is closed if the compliment of B is open.

Question 2

Interior point x

Answer 2

x is an interior point of A if there exists delta >0 st (x-delta,x+delta) is a subset A.

Question 3

Boundary point x

Answer 3

x is a boundary point of A if for all delta>0 (x-delta,x+delta) contains a point in A and a point not in A.

Question 4

Limit point x defn 1

Answer 4

If for all delta >0 (x-delta,x+delta) contains a point of A different then x.

Question 5

Closure

Answer 5

The closure of A is the set consisting of A and its limit points.

Question 6

Compact

Answer 6

A set B is compact if every open cover has a finite subcover.

Question 7

Sequence converges

Answer 7

The sequence {Xn} converges to the real number L if, for all epsolong >0, there exist and N in the natural numbers st. If n is in the natural numbers and n>N, then the absolute value of Xn-L

Question 8

Diverges

Answer 8

If {Xn} does not converge, then it diverges

Question 9

Bounded

Answer 9

A sequence is bounded if the terms of the sequence form a bounded set.

Question 10

Monotone increasing

Answer 10

A sequence is monotone increasing off an+1>= an for all n

Question 11

Monotone decreasing

Answer 11

A sequence is monotone increasing off an+1

Question 12

Cauchy-sequence

Answer 12

A sequence {Xn} is a Cauchy sequence if, given any epsolong >0, there exists N in the natural numbers st if n,m>N, then the absolute value of Xn-Xm

Question 13

Limit point x part 2

Answer 13

X is a limit point of a set of real numbers A if for all epsolong>0, (x-epsolong,x+ epsolong) contains infinitely many point of A.

Question 14

Limit of f as x approaches x knot is L

Answer 14

Let f be a function and x knot a limit point of the domain of f. The limit of f as x approaches x knot is L.

Question 15

f is continuous at x knot

Answer 15

Let f be a function and x knot is in the domain of f. Then we say f is continuous at x knot if given any epsolong>0, there exists a delta>0 st if the absolute value of x-x knot

Question 16

F is continuous on A

Answer 16

Let f be a function and A is the subset of the domain of f. F is continuous on A if, given any epsolong>0 and x knot is in A, there exists delta x knot, epsolong st if x is in domain of f and the absolute value of x - x knot

Question 17

Uniformly continuous

Answer 17

Let f be a function from a set A of the real numbers. F is uniformly continuous on A if, given epsolong > 0, there exists a delta epsolong > 0 st x,y in A and the absolute value of x-y

Question 18

Open

Answer 18

A set A of real numbers is open if for all x in A, there exists a delta x >0 st. (x - delta x, x +delta x) is a subset of A.