By John F. Randolph

This 1968 version covers the subsequent subject matters: Orientation; units and areas; sequences and sequence; degree and integration; degree idea; continuity; derivatives; and stieltjes integrals.

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Proof. Suppose [0, 1] is countable and let xu x2, · · · be such that [0, 1) = {xux2,x3,'··}, with Xi φ Xj, i Φ]. For each meJ select the decimal representation of xm which does not have all 9s from some place on; that is, xm — 0 . a m l a m 2

Thus Ä = A but E = Σ zfcwk. Then 0 < Σ |5z k - Awk\2 = Σ (Sz k - Awk)(Bzk - Awk) = Σ (Ezk - Awk)(Bzk - Awk) = Σ (BBzk zk - ABzk wk - ABzk wk + A2wk wk) = ßSA - ABB - ABB + ,42C = -A \B\2 + A2C = ^(AC - |ß| 2 ), which shows that if A > 0 then |2? | 2 ^ AC which is (12). If, however, A = 0 then for each k, \zk\ = 0, zk = 0, and (12) holds since all parts are zero. | 41 1-10 Complex Numbers The complex number z = x + z> (as the ordered pair (x, y) of real numbers) and the point labeled (x, y) of the Euclidean plane E2 are naturals to identify in what is called the Argand diagram (Fig.

A nonimaginary method by which complex numbers could have been introduced is by means of ordered pairs of real numbers. Definition 1. An ordered pair (x, y) of real numbers is defined to be a complex number z. Equality, addition, and multiplication are extended to complex numbers *i = (*ι> ^ι) and z2 = (x2, y2) by (i) z1 = z2 iff *! = x2 and y± = y2 . (ii) zx+z2 = (xi + x2, yt + y2). (iii) zxz2 = (xlx2 - yxy2, xty2 + x2 yj. 1-10 37 Complex Numbers Commutativity of the addition and multiplication operations for complex numbers then follows from the same laws for real numbers : zl + z2 = (x1 + x2,yi + y2) = O2 + *i> y 2 + yd = ^2 + zl9 and z±z2 = (x1x2 - yxy2, = (χ2χί xty2 + x2 yx) -y2yl9y2Xi + yi*2) = z2z1.