Inequalities like a ≤ x ≤ b, where a and b are arbitrary real numbers such that a < b, define intervals on the real line. The inequality above defines an interval that includes its extreme points. In such a case, we use the notation [a, b] to denote the interval, and we speak of a closed interval. On the contrary, inequalities a < x < b define the open interval (a, b). For instance, the point x = 10 belongs to the closed interval [0, 10] but does not belong to open interval (0, 10). We may also consider open-closed intervals like [a, b), corresponding to inequalities a ≤ x < b, or (a, b], corresponding to inequalities a < x ≤ b.
If both a and b are finite, we have a bounded interval. We may also consider unbounded intervals by extending the real line to include ±∞. Examples of unbounded intervals are
- (−∞, a], corresponding to x ≤ a
- [a, +∞), corresponding to x ≥ a
- (−∞, +∞), which is the whole real line
Note that there are infinite integer numbers in 
, as well as infinite real numbers in 
. However, we cannot say that the two sets have the same “order of infinity.” To see this informally, note that there are infinite real numbers even in a bounded interval like [0, 1]. To get an infinite set of integer numbers, we have to consider an unbounded interval.
Whenever we have an infinite collection of items that can be counted, i.e., can be placed in correspondence with the set of integer numbers, we speak of a countably infinite set. For instance, a sequence of arbitrary real numbers xk for k = 1, 2, 3, …, is an infinite sequence, but it includes less points than the bounded interval [0, 1]. More generally, we speak of denumerable or countable sets, including also finite collections of elements.
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