Supremum and infimum: characterization with ε
Supremum and Infimum of sequences
A sequence is bounded, if the set range {a_n : n } is a bounded set.
We define:
1) greatest lower bound or infimum of A, denoted by inf A := T, if
- T ≤ A, i.e., T is a lower bound and
- x ≤ A ⇒ x ≤ T, i.e., there is no greater lower bound.
- A ≤ T, i.e., T is an upper bound and
- A ≤ x ⇒ T ≤ x, i.e., there is no smaller upper bound.
- T ≤ A, i.e., T is a lower bound and
- ∀ε > 0 ∃a ∈ A: a < T + ε, i.e., T comes arbitrarily close to A.
- if A ≤ T, i.e., T is an upper bound and
- ∀ε > 0 ∃a ∈ A: a > T − ε, i.e., T comes arbitrarily close to A