Definitions Midterm 2

Question 1

condtitions satisfying least upper bound propert

Answer 1

any subset S c F that is non-empty and bounded from above has a lub

Question 2

conditions satisfying greatest lower bound property

Answer 2

any subset S c F that is non-empty and bounded from below has a glb

Question 3

convergence

Answer 3

A sequence (a_n)_n of real numbers converges to the limit a if, for all E >0, there is N c N s.t. |a_n-a|< E for all n ≥ N

Question 4

triangle inequality

Answer 4

If a,b c R, then, then |a+b| ≤ |a| + |b|

Question 5

open set

Answer 5

S c R is open if for all x c S there exists

E>0 s.t. (x-E, x+E) c S

Question 6

dense set

Answer 6

A subset S c R is dense is for every a, b c R, there exists s c S with a

Question 7

Archimedean Property

Answer 7

If a and b are positive real umbers, then there exists a natural number n such that na > b

For any x within R, there exists n > x

Question 8

lub (s)

Answer 8

• lub (S) ≥ s for all s c S
• if E>0, then there exists an s c S s.t. s> lub (s) - E