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## 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 “$$\ge$$”:

1. $$D$$ 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 $$D$$.
2. $$D$$ 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]

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