![]() It’s a fact that every Cauchy sequence converges to a real number as its limit, which means that every Cauchy sequence defines a real number (its limit). Such sequences are called Cauchy sequences. To be precise, giving any number as small as you like, you can be sure that if you move far enough along the sequence, all terms to follow are within of each other.Ĭauchy sequences are named after the French mathematician Augustin Louis Cauchy, 1789-1857. This is why, in this new way of thinking about real numbers, we only consider those sequences whose terms get arbitrarily close together as you move further along the sequence. Moves off to infinity and doesn’t specify any real number. Not every sequence of rational numbers defines a real number, however. For example, the sequenceĬonverges to, trivially, so we can think of as being represented by this sequence. This works for rational numbers (fractions and integers) too. To fractions, we can think of as defined by the sequence So, ditching decimal expansions altogether and converting the sequence of approximations for , Thus, implicit in the concept of decimal expansions is another way of looking at the real numbers: rather than representing them by their decimal expansions, we represent them by sequences of rational numbers that converge to them. The same can be said for any other irrational number and its decimal expansion. ![]() The sequence converges to in its limit: its terms get closer and closer to and we can get arbitrarily close to by going far enough down the sequence. For all practical purposes, the infinite decimal expansion of gives us a sequence of rational numbers that give better and better approximations to. The numbers in this approximating sequence have finite decimal expansions, so they are rational numbers. ![]()
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