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

Upper bound

Let \( E \) be a subset of \( \mathbb{R} \), and let \( M \) be a real number. We say that \( M \) is an upper bound for \( E \) if \( M \ge x \) for every element \( x \) in \( E \).

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

Example

The interval \( E:= \{ x \in R : 0 \le x \le 1 \} \) has 1 as an upper bound. All numbers greater than 1 are also upper bounds. 

The set of positive reals, \( \mathbb{R}^+ \), 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). 


Source

Tao, Analysis I