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# Which is a member of s and does not coincide with ξ

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, which is a member of S and does not coincide with ξ . Obviously we may repeat this argument, with ξ 2 in the place of ξ 1 ; and so on indefinitely. In this way we can determine as many points ξ 1 , ξ 2 , ξ 3 , . . . as we please, all belonging to S , and all lying inside the interval [ ξ - δ, ξ + δ ]. A point of accumulation of S may or may not be itself a point of S . The examples which follow illustrate the various possibilities. Examples IX. 1. If S consists of the points corresponding to the posi- tive integers, or all the integers, there are no points of accumulation. 2. If S consists of all the rational points, every point of the line is a point of accumulation. * This clause is of course unnecessary if ξ does not itself belong to S .

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[I : 19] REAL VARIABLES 34 3. If S consists of the points 1, 1 2 , 1 3 , . . . , there is one point of accumulation, viz. the origin. 4. If S consists of all the positive rational points, the points of accumulation are the origin and all positive points of the line. 19. Weierstrass’s Theorem. The general theory of sets of points is of the utmost interest and importance in the higher branches of analysis; but it is for the most part too difficult to be included in a book such as this. There is however one fundamental theorem which is easily deduced from Dedekind’s Theorem and which we shall require later. Theorem. If a set S contains infinitely many points, and is entirely situated in an interval [ α, β ] , then at least one point of the interval is a point of accumulation of S . We divide the points of the line Λ into two classes in the following man- ner. The point P belongs to L if there are an infinity of points of S to the right of P , and to R in the contrary case. Then it is evident that conditions (i) and (iii) of Dedekind’s Theorem are satisfied; and since α belongs to L and β to R , condition (ii) is satisfied also. Hence there is a point ξ such that, however small be δ , ξ - δ belongs to L and ξ + δ to R , so that the interval [ ξ - δ, ξ + δ ] contains an infinity of points of S . Hence ξ is a point of accumulation of S . This point may of course coincide with α or β , as for instance when α = 0, β = 1, and S consists of the points 1, 1 2 , 1 3 , . . . . In this case 0 is the sole point of accumulation. MISCELLANEOUS EXAMPLES ON CHAPTER I. 1. What are the conditions that ax + by + cz = 0, (1) for all values of x , y , z ; (2) for all values of x , y , z subject to αx + βy + γz = 0; (3) for all values of x , y , z subject to both αx + βy + γz = 0 and Ax + By + Cz = 0? 2. Any positive rational number can be expressed in one and only one way in the form a 1 + a 2 1 · 2 + a 3 1 · 2 · 3 + · · · + a k 1 · 2 · 3 . . . k ,
[I : 19] REAL VARIABLES 35 where a 1 , a 2 , . . . , a k are integers, and 0 5 a 1 , 0 5 a 2 < 2 , 0 5 a 3 < 3 , . . . 0 < a k < k. 3. Any positive rational number can be expressed in one and one way only as a simple continued fraction a 1 + 1 a 2 + 1 a 3 + 1 · · · + 1 a n , where a 1 , a 2 , . . . are positive integers, of which the first only may be zero.

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