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By A. Harnack

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N! N! 2 Here lim xn = lim zn = 0, and by the Comparison Test, the limit sought is zero: 0 = xn < n→∞ n→∞ lim n→∞ an = 0. n! 3. A sequence {xn } is called, respectively, increasing, nondecreasing, decreasing, or nonincreasing if xn+1 > xn , xn+1 xn , xn+1 < xn , or xn+1 xn , n = 1, 2, 3, . . A sequence that is either nondecreasing or nonincreasing is called monotone, and a sequence that is increasing or decreasing is called strictly monotone. It is clear that a strictly monotone sequence is monotone, but not conversely.

Obviously, the subtraction operation is defined as (a, b) − (a1, b1 ) = (a − a1, b − b1). Then with the properties (1)–(5), the set of complex numbers forms a ring. If we define the division operation in this ring as follows, (6) aa1 + bb1 a1 b − ab2 (a, b) = , 2 , (a1 , b1 ) = (0, 0), (a1 , b1 ) a21 + b21 a1 + b21 then we have that the set of complex numbers forms a field. 18 Single Variable Differential and Integral Calculus Moreover, for any real number we can write (a1 , 0) + (a2 , 0) = (a1 + a2, 0), (a1 , 0) · (a2 , 0) = (a1 a2 , 0).

Inductively, we obtain a sequence subintervals and choose one of them, A2 with x2 ∈ of closed intervals {An } such that for all n (n = 1, 2, . ) A ⊃ A1 ⊃ A2 ⊃ · · · ⊃ An ⊃ · · · 1 and so tend to zero for 3n infinitely large n. By Cantor’s Theorem, there is one and only one point ζ common to all and xn ∈ / An . Obviously, the lengths of the subintervals An are nested intervals An . But since ζ is in A1 , ζ must be in A. But the sequence {xn } does not contain the point ζ . Hence the set of points A cannot be represented by the sequence {xn }.

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An Intro to the Study of the Elements of the Diff and Int Calculus by A. Harnack


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