**Definition**: Bounded Subsets of Real Numbers

Let \(D\) be a non-empty subset of real numbers. Using the definition of the order relation for real numbers “\(\ge\)”:

- \(D\) is called
**bounded above**, if there is a real number \(B\) with \(x \le B\) for all \(x\in D\). In this case, \(B\) is called an**upper bound**of \(D\). - \(D\) is called
**bounded below**, if there is a real number \(B\) with \(x \ge B\) for all \(x\in D\). In this case, \(B\) is called a**lower bound**of \(D\).

If \(D\) is bounded above and bounded below, or if \(|x|\le |B|\) for all \(x\in D\), then we say that \(D\) is **bounded**.

| | | | | created: 2014-04-26 22:23:37 | modified: 2018-01-02 20:34:01 | by: *bookofproofs* | references: [581]

## 1.**Corollary**: A Criterion for Subsets of Real Numbers to be Bounded

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[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen”, Vieweg Studium, 1983

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