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Proof of Proposition 2.14
Proposition 2.14.
Let C*N be the greatest fixed
point of fE. Then
C*N(E) =
K*N(E).
Proof.
We have shown that
K*N(E) is a
fixed point of fE, so we only need to
show that
K*N(E) is
the greatest fixed point. Let B be a fixed point of
fB. We want to show that B
KkN(E)
for each value
k
1.
We will proceed by induction on k. By hypothesis,
B = fE(B) =
K1N(E
B)
K1N(E)
by monotonicity, so we have the k=1 case. Now suppose that for
k=m, B
KmN(E).
Then by monotonicity,
| (i) |
K1N(B)
K1NK
mN(E) =
Km+1N(E) |
We alo have:
| (ii) |
B =
K1N(EB)
K1N(B) |
by monotonicity, so combining (i) and (ii) we have:
B
K1N(B)
Km+1N(E)
completing the induction.
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Stanford Encyclopedia of Philosophy