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		provides a particular example of this. Chen[36] proved the analogous result for the Bcn criteria.

		Terminology: We say that x is the point of convergence of M on f if and only if and x is the least number such that, for all y > x, M(f[x]) = M(f[y]).

		7.21 Proposition Suppose M is a scientist and .

		(a) There is an such that either or else and, if x is the point of convergence of M on f, then for each , .

		(b) There are infinitely many such that either or else and, if x is the point of convergence of M on f, then for each , .

		Part (a) of this proposition follows from the proof of Proposition 6.5. We leave it to the reader to check the details of this. Part (b) follows as an easy consequence of part (a). Using this proposition we can establish the relation of the Huex criteria to the Exn criteria and some final details of the relation of the Huex with the Uex and Aex criteria.

		7.22 Proposition Suppose a, and m, .

		(a) If , then .

		(b) If n > a · (m + 1), then .

		(c) If a < 1, then .

		(d) If a< 1 and , then .

		(e) If a < b, then .

		Proof: Observation: If , then can have as many as [a · (m + 1)] members in any interval of length m, but no more than that.

		We have from the Observation that, if and if , then ; hence part (a) follows.

		Part (b) follows from the Observation and Proposition 7.21(a).

		Part (c) follows from Proposition 7.17(c) and Proposition 7.20.

		By Definitions 7.16 and 7.18, if , then . By part (c), this containment is strict; hence part (d) follows.

		Suppose a < b and suppose q and r are rationals such that a < q < r < b. The S of the proof of Lemma 7.9 is clearly in Huexr, hence by the proof of Lemma 7.9,