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A_k = {1, a_2, ... a_k} is an h-basis for X if every positive integer notexceeding X can be expressed as the sum of no more than h values a_i; X(h) iscalled the h-range of the basis. h_0 is the smallest value of h for which X(h)is greater than or equal to a_k, and h_1 is the smallest value for which X(h+1)= X(h) + a_k for all h greater than or equal to h_1. h_2 identifies a further"stabilisation" in the h-range - a definition is included in the body of thepaper. It is known that h_1 and h_2 do not exceed h_0 for h-bases A_3, butpublished proofs are complicated (see Ch. VIII of [4], where references [3] and[5] are given). This paper introduces the concept of a "stride generator" A ={1, a_2, a_3} which, while sharing some of the properties of a basis A_3, issimpler to treat mathematically. We establish a relationship between stridegenerators and h-bases, and show that the result follows immediately if thestride generator underlying a basis has a particular property - here called"canonicality". The proof is lengthy (with a number of special cases toconsider), but the underlying principles remain simple.

A proof that h_1, h_2 <= h_0 for any h-basis A_3