#### Supplement to Common Knowledge

## 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
**K**^{*}_{N}(*E*) is a
fixed point of *f*_{E}, so we only need to show
that **K**^{*}_{N}(*E*) is
the greatest fixed point. Let *B* be a fixed point of
*f*_{B}. We want to show that *B* ⊆
**K**^{k}_{N}(*E*)
for each value *k*≥1. We will proceed by induction on
*k*. By hypothesis,

B=f_{E}(B) =K^{1}_{N}(E∩B) ⊆K^{1}_{N}(E)

by monotonicity, so we have the *k*=1 case. Now suppose that for
*k*=*m*, *B* ⊆
**K**^{m}_{N}(*E*).
Then by monotonicity,

(i) K^{1}_{N}(B) ⊆K^{1}_{N}K^{m}_{N}(E) =K^{m+1}_{N}(E)

We also have:

(ii) B=K^{1}_{N}(E∩B) ⊆K^{1}_{N}(B)

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

B⊆K^{1}_{N}(B) ⊆K^{m+1}_{N}(E)

completing the induction. □