Proof of Proposition 2.18

Proposition 2.18.
Let $C^*_N$ be the greatest fixed point of $f_E$. Then $C^*_N (E) = K^*_N (E)$.

Proof.
We have shown that $\mathbf{K}^*_N (E)$ is a fixed point of $f_E$, so we only need to show that $\mathbf{K}^*_N(E)$ is the greatest fixed point. Let $B$ be a fixed point of $f_B$. We want to show that $B \subseteq \mathbf{K}^k_N(E)$ for each value $k\ge 1$. We will proceed by induction on $k$. By hypothesis,

$$B = f_E (B) = \mathbf{K}^1_N (E\cap B) \subseteq \mathbf{K}^1_N (E)$$

by monotonicity, so we have the $k=1$ case. Now suppose that for $k=m,$ $B \subseteq \mathbf{K}^m_N(E).$ Then by monotonicity,

$$\tag{i} \mathbf{K}^1_N(B) \subseteq \mathbf{K}^1_N \mathbf{K}^m_N (E) = \mathbf{K}^{m+1}_N(E)$$

We also have:

$$\tag{ii} B = \mathbf{K}^1_N (E\cap B) \subseteq \mathbf{K}^1_N (B)$$

by monotonicity, so combining (i) and (ii) we have:

$$B \subseteq \mathbf{K}^1_N (B) \subseteq \mathbf{K}^{m+1}_{N}(E)$$

completing the induction. $\Box$

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