Vector space/Set of sequences in ordered K/Cauchy-sequences as linear subspace/Exercise

Let ${}K$ denote an ordered field, and set

${}C={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{Cauchy sequence in }}K\right\}}\,.$

Show that ${}C$ is a linear subspace of the space of sequences

${}F={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{sequence in }}K\right\}}\,.$

Create a solution