#### Stanford Encyclopedia of Philosophy

Supplement to Common Knowledge

## Proof of Proposition 2.14

**Proposition 2.14**.

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 alo 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.

Copyright © 2002 by

**Peter Vanderschraaf**

*peterv@cyrus.andrew.cmu.edu*
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*First published: June 12, 2002*

*Content last modified: June 12, 2002*