Least upper, greatest lower bound. Theorems about sup and inf

Definition

Numerical set Х is called bounded from above, if there exists number M such that x ≤ M, ∀ x ∈ X.

Definition

Numerical set Х is called bounded from below, if there exists number m such that m ≤ x, ∀ x ∈ X.

Definition

Number is called least upper bound of the set Х, if

(1) ,   (2) .

Notation: = sup X

Definition

Number is called greatest lower of the set Х, if

(1) ,   (2) .

Notation: = inf X