Bounded Sets and Bounded Functions in a Metric Space

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Definition: Let be a metric space. A subset is said to be Bounded if there exists a positive real number such that for some . The set is said to be Unbounded if it is not bounded.

By the definition above, we see that is bounded if there exists some open ball with a finite radius that contains .

For example, consider the metric space where is the discrete metric defined for all by:

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Let . Then by the definition of the discrete metric, for all we have that . Therefore, if we consider any point and take then:

Therefore, is bounded. This shows that every subset of is bounded with respect to the discrete metric. In fact, the wholeset is also bounded and for any .

For another example, consider the metric space where is the Euclidean metric. Consider the following set:

The set above is the first octant of , and is actually unbounded. To prove this, suppose that instead is bounded. Then there exists a maximal distance between some pair of points , say:

Then . For consider the point . Then:

Since for each we see that:

But then implies which is a contradiction. Therefore our assumption that was bounded is false.