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- TL;DR Summary
- On the real numbers, a set ##D## is bounded iff there exists a positive ##M## such that for all ##y\in D##, ##|y|\leq M##.

On the real numbers, a set ##D## is unbounded iff for all positive ##M##, there is a ##y\in## such that ##|y|>M##.

I read in my textbook

So my question is if the following definition of unboundedness is valid: "A set ##A## is unbounded if for all n-rectangles ##D##, there is a ##y\in A## such that there is an open set containing ##y## that is a subset of ##\mathbb{R}^n \cap D^c##".

*Calculus on Manifolds*by Spivak that a set ##A\subset \mathbb{R}^n## is bounded iff there is a closed n-rectangle ##D## such that ##A\subset D##. It should be plain that if I wanted to define unboundedness, I should just say something along the lines of: "A set ##A\subset \mathbb{R}^n##is unbounded iff for all n-rectangle ##D##, ##A## is not contained in ##D##.DSo my question is if the following definition of unboundedness is valid: "A set ##A## is unbounded if for all n-rectangles ##D##, there is a ##y\in A## such that there is an open set containing ##y## that is a subset of ##\mathbb{R}^n \cap D^c##".