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

Theorem

Numerical set has unique sup and inf.

Proof

Let it be two: M₁ and M₂, M₁ < M₂. Then for ε = (M₂ − M₁)/2 there exists .

The last contradicts condition that M₁ — the upper bound.

Theorem

Every bounded from above numerical set X has the least upper bound.

Proof

Let Y be the set of upper bounds for X. It is not empty. Elements of these sets are connected by inequality: x ≤ y. According to axiom 5 there exists ∈ R , for which . Number y = − ε ∉ Y, that is there exists , for which − ε.