As an alternative to classical calculus, Grossman and Katz (Non-Newtonian Calculus, 1972) introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and bigeometric calculus etc. Following Grossman and Katz, we construct the field R(N) of non-Newtonian real numbers and the concept of non-Newtonian metric. Also, we give the triangle and Minkowski's inequalities in the sense of non-Newtonian calculus. Later, we respectively define the sets ω(N), ∞ (N), c(N), c 0

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... ) and p (N) of all, bounded, convergent, null and p-absolutely summable sequences in the sense of non-Newtonian calculus and show that each of the sets forms a vector space on the field R(N) and a complete metric space. MSC: 26A06; 11U10; 08A05