Convergent Sequences are Cauchy

Last updated Nov 1, 2022

Let $(M, d)$ be a Metric Space and let $({x}{n}){n=1}^{\infty} \subset M$. If $x_{n} \to x \in M$, then $({x}{n}){n=1}^{\infty}$ is a Cauchy Sequence.

Let $\epsilon > 0$. Since $x_{n} \to x$, we can find $N \in \mathbb{N}$ so that $\forall n \geq N$, $d(x_{n}, x) < \frac{\epsilon}{2}$. Then $\forall n,m \geq N$, $$d(x_{n},x_{m}) \leq d(x_{n}, x) + d(x, x_{m}) < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon,$$ so $({x}{n}){n=1}^{\infty}$ is a Cauchy Sequence.

• Sequence
• Sequence Convergence
• Natural Numbers
• Real Numbers