PomeLemma 19.7.18.7887065F3C7C The above defined $d_{\mathrm{sg}}$ is a pseudometric on $\hat{\mathcal{S}}(C)$.
Lemma 19.7.187887065F3C7C
Proof. Only the triangle inequality needs to be argued. Take $S,S',S''\in \hat{\mathcal{S}}(C)$. We claim that for any $k\in \mathbb {N}$,
\[ |S_ k|+|S'_ k|-2|S_ k\cap S'_ k|+|S''_ k|+|S'_ k|-2|S''_ k\cap S'_ k|\geq |S_ k|+|S''_ k|-2|S_ k\cap S''_ k|. \]
From this the triangle inequality follows. To argue the claim, we rearrange it to the following form:
\[ |S'_ k|-|S_ k\cap S'_ k| \geq |S'_ k\cap S''_ k|-|S_ k\cap S''_ k|, \]
which is obvious. $\square$
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