Understanding Bounded Intervals in Mathematics: Definition and Examples

bounded intervals

both of its endpoints are real numbers (finite)

In mathematics, a bounded interval is a set of real numbers that is defined by a starting point and an ending point. The starting point is called the lower bound, and the ending point is called the upper bound. A bounded interval includes all the real numbers between the lower bound and upper bound, as well as the lower and upper bounds themselves.

For example, the interval [0,1] is a bounded interval because it includes all real numbers between 0 and 1, including 0 and 1 themselves. Another example is the interval [−2,4], which includes all real numbers between −2 and 4, including −2 and 4.

It is important to note that a bounded interval has a finite size. In other words, there is a finite distance between the upper and lower bounds. This is in contrast to an unbounded interval, which has either no upper bound, no lower bound, or both.

Bounded intervals are used in a variety of mathematical contexts, including calculus, real analysis, and number theory. They are important because they allow mathematicians to rigorously define sets of real numbers that have certain properties, which can then be used to prove theorems and solve problems.

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