Cauchy Sequences in Metric Spaces

# Cauchy Sequences in Metric Spaces

Just like with Cauchy sequences of real numbers - we can also describe Cauchy sequences of elements from a metric space $(M, d)$.

 Definition: Let $(M, d)$ be a metric space. A sequence $(x_n)_{n=1}^{\infty}$ is said to be a Cauchy Sequence if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ then $d(x_m, x_n) < \epsilon$.

Our first result on Cauchy sequences tells us that all convergent sequences in a metric space are Cauchy sequences.

 Theorem: Let $(M, d)$ be a metric space and let $(x_n)_{n=1}^{\infty}$ be a convergent sequence such that $\lim_{n \to \infty} x_n = p$. Then $(x_n)_{n=1}^{\infty}$ is also a Cauchy sequence.
• Proof: Let $(x_n)_{n=1}^{\infty}$ be a convergent sequence such that $\lim_{n \to \infty} x_n = p$ and let $\epsilon > 0$ be given. Then for $\epsilon_1 = \frac{\epsilon}{2}$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then:
(1)
\begin{align} \quad d(x_n, p) < \epsilon_1 = \frac{\epsilon}{2} \end{align}
• By applying the triangle inequality, we have that for all $m, n \geq N$ we have that:
(2)
\begin{align} \quad d(x_m, x_n) \leq d(x_m, p) + d(x_n, p) \leq \epsilon_1 + \epsilon_1 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}
• Hence $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence.