Math and science::Analysis::Tao::05. The real numbers

Uniqueness of least upper bound, proposition

Let

E

be a subset of

R

. Then

E

can have at most one least upper bound.

This is a simple proposition.

Proof

Let

E

be a subset of

R

, and let

M

and

M^{'}

be two upper bounds of

E

. If both upper bounds are least upper bounds then both

M \leq M^{'}

and

M^{'} \leq M

, by the definition of least upper bounds. This implies that

M = M^{'}

and that there is only a single least upper bound.

Upper bound def → least upper bound def→ uniqueness of least upper bound → existence of least upper bound → supremum def

Tao, Analysis I

26.08.2019