Write formally: "A set of all Cauchy sequences"

Question

Write formally: "A set of all Cauchy sequences"
My attempt at the solution:
$$C = \{a_n : \forall\epsilon \in \mathbb{R} \land\epsilon >0 \exists m\in \mathbb{N} |a_n -a_m| < \epsilon\}$$ Is is correct or I'm missing something? Also, does it matter if I write $a_n - a_m$ or $a_m - a_n$ inside the absolute value? This will be the same in terms of numbers but I am not sure if it will be correct in terms of a formal definition.

Answer 1

You definition of $C$ does not correspond to what a Cauchy sequence is. A sequence $(a_n)_{n\in\mathbb{N}}\subset \mathbb{R}$ is said to be Cauchy if

$$\left(\forall \epsilon >0\right)\, \left(\exists n_0\in\mathbb{N}\right)\,\left(\forall m,n\in\mathbb{N}\text{ with $m,n\geq n_0$}\right)\, |a_m-a_n|<\epsilon.$$

So, the inequality must be valid for all pairs of indices after $n_0$.

That said, after you've defined what a Cauchy sequence is, you can just say

$$C=\{(a_n)_{n\in\mathbb{N}}\,|\, \text{ $(a_n)_{n\in\mathbb{N}}$ is Cauchy}\}$$

and this is perfectly fine.