In 'Analysis' by Elliott H. Lieb and Michael Loss, Theorem 2.13 is the Mazur's lemma for $L^{p}(\Omega)$ ($1<p< \infty$). 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.'


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