A Cauchy sequence is a sequence of real numbers that converges to a limit, meaning that as the terms in the sequence get closer and closer to each other, the sequence gets closer to a single value. This concept was first introduced by French mathematician Augustin-Louis Cauchy in the 19th century.
A Cauchy sequence is formally defined as a sequence {xn} of real numbers such that for any positive real number ε, there exists a positive integer N such that for all n, m ≥ N, the absolute value |xn - xm| < ε. In other words, the terms in the sequence become arbitrarily close to each other as the index approaches infinity.
A Cauchy sequence is a type of convergent sequence, meaning it approaches a limit as the index approaches infinity. However, a Cauchy sequence differs from a general convergent sequence in that its terms become arbitrarily close to each other, rather than just approaching a single value.
No, a Cauchy sequence can only have one limit. This is because the definition of a Cauchy sequence requires the terms to become arbitrarily close to each other, meaning that as the index approaches infinity, the terms will approach a single value.
Cauchy sequences are important in mathematics because they provide a rigorous foundation for the concept of convergence. They are used in various areas of mathematics, such as calculus, analysis, and number theory, to prove the existence of limits and to define important concepts such as continuity and differentiability.