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

Least upper bound

Let E be a subset of 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∈R:0≤x≤1} has 1 as a least upper bound. 


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

Source

Tao, Analysis I