Extended real number line

Real numbers with +∞ and −∞ added

In mathematics, the extended real number system is obtained from the real number system by adding two elements denoted and that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities. For example, the infinite sequence of the natural numbers increases infinitively and has no upper bound in the real number system ; in the extended real number line, the sequence has as its least upper bound and as its limit. In calculus and mathematical analysis, the use of and as actual limits extends significantly the possible computations. It is the Dedekind–MacNeille completion of the real numbers.

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