A sequence is bounded, if the set range {a_n : n [math]\in N[/math] } [math][/math]is a bounded set. [br][br]We define: [br]1) greatest lower bound or infimum of A, denoted by inf A := T, if[br][list][*]T ≤ A, i.e., T is a lower bound and[/*][*]x ≤ A ⇒ x ≤ T, i.e., there is no greater lower bound.[/*][/list]1) least upper bound or supremum of A, denoted by sup A := T, if[br][list][*]A ≤ T, i.e., T is an upper bound and[/*][*]A ≤ x ⇒ T ≤ x, i.e., there is no smaller upper bound.[/*][/list][br]We give equivalent definition of supremum and infimum: [br](Although it looks more complicated at first sight, this formulation is sometimes very helpful).[br]Let $A ⊂ R$ be bounded from below. Then: [br]1) inf A:= T if and only if[br][list][*] 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.[/*][/list]2) sup A := T if and only [br][list][*]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[/*][/list]