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

Upper bound

Let

E

be a subset of

R

, and let

M

be a real number. We say that

M

is an upper bound for

E

M \geq x

for every element

x

E

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

The interval

E := {x \in R : 0 \leq x \leq 1}

has 1 as an upper bound. All numbers greater than 1 are also upper bounds.

The set of positive reals,

R^{+}

, has no upper bound.

The empty set,

\emptyset

, has every real as an upper bound. This is true as any real is greater than all elements of the empty set (vacously true, but still true).

Tao, Analysis I

26.08.2019