Algorithmic combinatorics by Even S.

By Even S.

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We have known the following result in metric spaces. 1 Any sequence {xn } in a metric space has at most one limit point. For x0 ∈ M and ǫ > 0, a ǫ-disk about x0 is defined by B(x0 , ǫ) = { x | x ∈ M, ρ(x, x0 ) < ǫ}. If A ⊂ M and there is an ǫ-disk B(x0 , ǫ) ⊃ A, we say A is a bounded point set of M. 2 Any convergent sequence {xn } in a metric space is a bounded point set. Now let (M, ρ) be a metric space and {xn } a sequence in M. If for any number ǫ > 0, ǫ ∈ R, there is an integer N such that n, m ≥ N implies ρ(xn , xm ) < ǫ, we call {xn } a Cauchy sequence.

By definition, we know that for any integer n, n ≥ 1, there exists an integer i, 1 ≤ i ≤ m such that xn , yn ∈ Mi . Whence, we inductively get that 0 ≤ ρi (xn , yn ) ≤ αn ρ1 (x0 , y0). Notice that 0 < α < 1, we know that lim αn = 0. Thereby there exists an n→+∞ integer i0 such that ρi0 (lim xn , lim yn ) = 0. n n Therefore, there exists an integer N1 such that xn , yn ∈ Mi0 if n ≥ N1 . Now if n ≥ N1 , we get that ρi0 (xn+1 , xn ) = ρi0 (T xn , T xn−1 ) ≤ αρi0 (xn , xn−1 ) = αρi0 (T xn−1 , T xn−2 ) ≤ α2 ρi0 (xn−1 , xn−2 ) ≤ · · · ≤ αn−N1 ρi0 (xN1 +1 , xN1 ).

For a multi-ring R = m i=1 Ri , let S ⊂ R and O(S) ⊂ O(R), if S is also a multi- ring with a double operation set O(S) , then we call S a sub-multi-ring of R. We get a criterion for sub-multi-rings in the following. 8 For a multi-ring R = m i=1 a sub-multi-ring of R if and only if (S S Ri , a subset S ⊂ R with O(S) ⊂ O(R) is Rk ; +k , ×k ) is a subring of (Rk ; +k , ×k ) or Rk = ∅ for any integer k, 1 ≤ k ≤ m. Proof For any integer k, 1 ≤ k ≤ m, if (S (Rk ; +k , ×k ) or S Rk = ∅, then since S = m (S i=1 Rk ; +k , ×k ) is a subring of Ri ), we know that S is a sub- multi-ring by the definition of a sub-multi-ring.

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