Supremum and infimum: characterization with ε

Autor:
Lucia Del Chicca, Andreas Lindner
Thema:
Analysis

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.
1) least upper bound or supremum of A, denoted by sup A := T, if
  • A ≤ T, i.e., T is an upper bound and
  • A ≤ x ⇒ T ≤ x, i.e., there is no smaller upper bound.
We give equivalent definition of supremum and infimum: (Although it looks more complicated at first sight, this formulation is sometimes very helpful). Let $A ⊂ R$ be bounded from below. Then: 1) inf A:= T if and only if
  • 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.
2) sup A := T if and only
  • 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

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