Math 521 Flashcards

ordered field

you can add, subtract, divide, multiply and you can tell that x<y for some x,y in Q

how do we get R

we start with Q and build a bigger ordered field around R

triangle inequality

∀ x,y in R, |x+y| ≤ |x|+|y|

triangle inequality corollary

look up

Let E be a subset of R, E is bounded above if

∀x∈E, x≤β, for some β∈R

Let E be a subset of R, E is bounded below if

∀x∈E, α≤x, for some α∈R

Let E be a subset of R, E is bounded (words only intuition)

if E is bounded above AND below

Let E be a subset of R, E is bounded (math language)

iff ∃m≥0 s.t. ∀x∈E, |x|≤m

definition of absolute value

if x∈R, then |x| = { [x if x>0] [-x if x<0] }

∀x,y ∈R, |xy|=

|x||y|

∀x∈R

-|x| ≤ x ≤ |x|

|a|≤b IFF

-b≤a≤b

Q has holes theorem

add from lecture 8/19

axiom of completeness

every nonempty set of R that is bounded above, has a supremum (least upper bound) in R

R has AOC but

Q does not

lower bound aoc

8/21 pg2

archimedean principle intuition

you can always make a number bigger by multiplying

archimedean principle (math language)

∀x>0, y ∈ R, ∃n∈N s.t. nx>y

density of Q

8/24