Darboux ’ s Theorem Noboru

In this article, we have proved the Darboux's theorem. This theorem is important to prove the Riemann integrability. We can replace an upper bound and a lower bound of a function which is the definition of Riemann integration with convergence of sequence by Darboux's theorem. We adopt the following rules: x, y are real numbers, i, j, k are natural numbers, and p, q are finite sequences of elements of R. We now state a number of propositions: (1) Let A be a closed-interval subset of R and D be an element of divs A. If vol(A) = 0, then there exists i such that i ∈ dom D and vol(divset(D, i)) > 0. (2) Let A be a closed-interval subset of R and D be an element of divs A. If x ∈ A, then there exists j such that j ∈ dom D and x ∈ divset(D, j). (3) Let A be a closed-interval subset of R and D 1 , D 2 be elements of divs A. Then there exists an element D of divs A such that D 1 ≤ D and D 2 ≤ D and rng D = rng D 1 ∪ rng D 2. (4) Let A be a closed-interval subset of R and D, D 1 be elements of divs A. Suppose δ (D 1) < min rng upper volume(χ A,A , D). Let given x, y, i. If i ∈ dom D 1 and x ∈ rng D ∩ divset(D 1 , i) and y ∈ rng D ∩ divset(D 1 , i), then x = y. (5) For all p, q such that rng p = rng q and p is increasing and q is increasing holds p = q. (6) Let A be a closed-interval subset of R and D, D 1 be elements of divs A. If D ≤ D 1 and i ∈ dom D and j ∈ dom D and i ≤ j, then indx(D 1 , D, i) ≤ indx(D 1 , D, j) and indx(D 1 , D, i) ∈ dom D 1. (7) Let A be a closed-interval subset of R and D, D 1 be elements of divs A. If D ≤ D 1 and i ∈ dom D and j ∈ dom D and i < j, then indx(D 1 , D, i) < indx(D 1 , D, j). (8) For …