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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?).

Tao, Analysis I