Boundedness Flashcards
Define “bounded above”
∃H∈ℝ such that ∀x∈S we have x≤H
Define “bounded below”
∃H∈ℝ such that ∀x∈S we have x≥H
Define “bounded”
Bounded above and bounded below
Give a condition for a real set S to be bounded.
∃H∈ℝ such that ∀x∈S we have |x|≤H
Define “unbounded above”
∀H∈ℝ ∃x∈S such that x>H
Define “unbounded below”
∀H∈ℝ ∃x∈S such that x
Define “maximum”
M is an upper bound of S, and M∈S
Define “minimum”
m is a lower bound of S, and m∈S
Theorem: Uniqueness of max/min
If they exist, max/min is unique
Define “infimum”
infS is the greatest number such that infS is a lower bound of S
Define “supremum”
supS is the lowest number such that supS is an upper bound of S
What is the condition for m=infS?
∀ε>0 ∃x∈S such that x
What is the condition for M=supS?
∀ε>0 ∃x∈S such that x>M-ε
What is the axiom of completeness?
Every non-empty real subset bounded above has a supremum. Every non-empty real subset bounded below has an infimum.
