## 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. $$\Box$$