Question 1

Question 1

Definition of bounded

Answer 1

A set S⊆R is bounded if and only if it is bounded above and below

Question 2

Definition of bounded above

Answer 2

A set S⊆R is bounded above if there exists c∈R such that x≤c for all x∈S. The number c is an upper bound for S

Question 3

Definition of bounded below

Answer 3

A set S⊆R is bounded below if there exists c∈R such that x≥c for all x∈S. The number c is a lower bound for S

Question 4

Definition of a maximum

Answer 4

Let S⊆R be bounded above. If there is an a∈R that:
i) is an upper bound for S: for all x∈S we have that x≤a
ii) a∈S

Question 5

Definition of a minimum

Answer 5

Let S⊆R be bounded below. If there is an a∈R that:
i) is a lower bound for S: for all x∈S we have that x≥a
ii) a∈S

Question 6

Definition of supremum

Answer 6

Let S⊆R be bounded above. Then a∈R is called the supremum of S if:
i) it is an upper bound for S: for all x∈S we have that x≤a
ii) it is smaller than any other upper bound: for all E>0 there exists x∈S such that x>a-E, a=sup S

Question 7

Definition of infimum

Answer 7

Let S⊆R be bounded below. Then a∈R is called the infimum of S if:
i) it is a lower bound for S: for all x∈S we have that x≥a
ii) it is larger than any other lower bound: for all E>0 there exists x∈S such that x<a+E, a=Inf S

Question 8

Definition of the completeness axiom (C1)

Answer 8

Every non-empty subset of R that is bounded above has a supremum/bounded below has a infimum.

Question 9

Archimedean principal

Answer 9

Let x∈R, then there exists n∈Z such that x<n

Question 10

Well-Ordering principal

Answer 10

Any non-empty subset S⊆Z which is bounded above has a maximum/bounded below has a minimum.

Question 11

If S is bounded and T is bounded is SnT bounded?

Answer 11

Yes, if c>0 such that | x|≤ c for all x∈T. Since SnT ⊆ T we also have | x|≤ c for all x∈SnT.