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

Least upper bound

Let \( E \) be a subset of \( \mathbb{R} \) and let \( M \) be a real number. We say that \( M \) is a least upper bound for E iff:
  1. \( M \) is an upper bound for \( E \).
  2. Any other upper bound for \( E \) is greater or equal to \( M \).

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 a least upper bound. 


The empty set does not have any least upper bound (why?).

Source

Tao, Analysis I