Cauchy sequence

A sequence over a metric space $(𝑋, 𝑑)$ is said to be Cauchy iff its terms get arbitrarily close together. More formally, a sequence $(𝑎 𝑛) \infty 𝑛 = 1$ is a Cauchy sequence iff. for every $𝜖 > 0$ there exists an integer $𝑁$ such that $𝑑 (𝑎 𝑚, 𝑎 𝑛) < 𝜖$ for any $𝑚 > 𝑛 > 𝑁$ . #m/def/anal

It is easy to prove that every convergent sequence is a Cauchy sequence using the triangle inequality. A metric space in which every convergent sequence is a Cauchy sequence is called a Complete metric space.

#state/tidy | #lang/en | #SemBr