Real Numbers

Question 1

Heine-Borel Theorem

Answer 1

S is compact if and only if it is closed and bounded

Question 2

Bolzano-Weierstrass Theorem

Answer 2

A bounded infinite set has at least one accumulation point

Question 3

Archimedean Property of R

Answer 3

A set N of natural numbers is unbounded above in R

Question 4

Archimedean Property Equivalents

Answer 4

1) For every z in R, there is a n in N such that n>z (there’s always a greater integer than any number) 2) For x>0 and y in R, there is an n in N such that nx>y (equivalently, n>y/x) 3) For each x>0 there is an n in N such that 0>1/n>x

Question 5

Density of Ir/rationals

Answer 5

If x and y are real numbers x<y>
</y>

Question 6

The intersection of ANY collection of closed sets

Answer 6

is closed

Question 7

The union of any FINITE collection of closed sets

Answer 7

is closed

Question 8

Definition of Compactness

Answer 8

S is covered by an infinite open cover and also by a finite sub cover of the open cover

Question 9

Triangle Inequality

Answer 9

|x+y|<=|x|+|y|

Question 10

Let s_n and a_n be sequences and let s be a real number, if for some k>0, m in the natural numbers and a_n→0

Answer 10

|s_n-s|<=k|a_n| for all n>=m, then s_n→s

Find an upper bound for numerator and lower for denominator

Question 11

Completeness Axiom

Answer 11

Every nonempty subset S of real numbers that is bounded above has a least upper bound. (sup S exists and is a real number)

1) rationals are not complete, i.e (0, sqrt(2)) the sup would be sqrt (2), but it is not in the set of Q rationals

Question 12

The union of any collection of open sets

Answer 12

is an open set

Question 13

The intersection of any finite collection of open sets

Answer 13

is an open set

Question 14

Boundedness

Answer 14

*Set S is said to bounded if it bounded above and bounded below

*it has a sup S and an inf S

Question 15

Neighborhood

N(x,epsilon)=

{y in R : |x-y| < epsilon}

Answer 15

Neighborhood x of radius ep is open interval (x-ep, x+ep)

Related: interior points

Question 16

Deleted Neighborhood

N*(x,epsilon)=

{y in R : 0 < |x-y| < epsilon}

Answer 16

(x-ep, x) U (x, x + ep)

Related: accumulation points

Question 17

Interior Point

Answer 17

A point is an interior point if there is A (at least one) N(x,ep) C S

Question 18

Boundary Point

Answer 18

If for EVERY neighborhood N(x,ep) the intersection of N with S is not empty and the intersection of N with S^c is not empty

Question 19

Open/Closed Sets

Answer 19

If bd S C S then S is closed

OR

if bd S C S^c then S is open

(MOST sets are neither/both)

Question 20

Accumulation Point

set of all accumulation points is S’

Answer 20

A point x in R is an acc pt if EVERY N*(x, ep) contains a point in S

Intersection of N*(x) and S is not empty

Question 21

Isolated Point

Answer 21

x in S but NOT in S’

it’s literally isolated (likely to be a bd since it’s neighborhoods will intersect with S^c)

Question 22

Closure

cl S = S U S’

Answer 22

S is closed iff S contains of its accumulation points

cl is closed set

S is closed iff S = cl S

cl S = S U bd S