# Cauchy sequences of Cauchy sequences are Cauchy sequences

In 'Analysis' by Elliott H. Lieb and Michael Loss, Theorem 2.13 is the Mazur's lemma for $$L^{p}(\Omega)$$ ($$1). Let $$f^{j}$$ be a sequence in $$L^{p}(\Omega)$$ that converges weakly to $$F$$. $$\tilde K$$ is defined to be the collection of all finite convex combinations of $$f^{j}$$. $$K$$ is defined to be $$\tilde K \cup \tilde K '$$ where $$\tilde K '$$ is the set of limit points of $$\tilde K$$.

The author says that one can prove $$K$$ is closed using the fact that 'Cauchy sequences of Cauchy sequences are Cauchy sequences.'

At a first glance, it seems to be awkward since a Cauchy sequence is indeed a Cauchy sequence. The author gives a hint that this is just an imitation of the construction of $$\mathbb{R}$$ from $$\mathbb{Q}$$.

Now I guess that it is somewhat related to the space of equivalence classes of Cauchy sequences in $$\tilde{K}$$. I mean if we denote such space to be $$\tilde{\tilde{K}}$$, then it would be just the closure of $$\tilde{K}$$ in $$L^{p}(\Omega)$$ by the completeness of $$L^{p}(\Omega)$$.

But still I don't understand why this can be explained by the phrase that 'Cauchy sequences of Cauchy sequences are Cauchy sequences.'