Vector space/Set of sequences/R/Cauchy sequences as subspace/Exercise

Let

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

be the set of all real Cauchy sequences. Show that ${}C$ is a linear subspace of the space of all sequences

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

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