TWO CONSTRUCTIONS OF THE REAL NUMBERS VIA ALTERNATING SERIES

Two further new methods are put forward for constructing the complete 
ordered field of real numbers out of the ordered field of rational numbers. The 
methods are motivated by some little known results on the representation of real 
numbers via alternating series of rational numbers. Amongst advantages of the 
methods are the facts that they do not require an arbitrary choice of "base" or 
equivalence classes or any similar constructs. The methods bear similarities to 
a method of construction due to Rieger, which utilises continued fractions.


INTRODUCTION.
The series of Enge] (1913) and Sylvester (1880) (see Perron [1])for represent:g real numbers have been studied in some detail.
Much less known is the fact that there are alternating series representations of real numbers in terms of rationals corresponding to the above.
The only references to these alternating series that we are aware of in the literature are in papers of E Remez [2] and H Salzer [3].
The series under discussion are as follows: Every real number A has a unique representation in the form A ao + i 1 + 1 + (-1)n+1 + (ao,a1,a 2 ).
a I a I a 2 a la2a3 a I a 2 a n are integers such that ai+1 > a i + 1 >_--2 for il.Furthermore.
say, where the A is rtionZ if and only if A has a finite representation (ao,a I (Compare this with the expansion of Engel (Perron [i]).) Corresponding to the series of Sylvester (Perron [1]) we have every real number A ao 1 1 + 1 + (-1)n+1 + a o,a1,a 2 )), a a 2 a 3 a n say, where the i are integers defined uniquely by A such that al>_--i and ai+ >_-ai(a i + 1) for i>--_1.Furthermore, A is rovt.Lonc if and only if A has a finite representation ((ao,a I an)).
In many ways, these representations may be compared with that by simple continued fractions.The main purpose of this note is to justify this remark by deriving some elementary properties of these alternating series representations and (with these results as an initial motivation) then developing two new methods for constructing the real number system from the ordered field of rational numbers.These methods are similar to one recently introduced by G J Rieger [4] for constructing the real numbers via continued fractions.The order relations in particular are defined in an analogous fashion.
The methods share with Rieger's method the advantage over other standard techniques that they do not require an arbitrary choice of a "base", or the use of (infinite) equivalence classes or similar such constructs.These properties are shared as well in the construction of real numbers using ordinary Sylvester and Engel series, considered in [5].Two important differences between those and the present methods are in the definition of the order relations for the series, as well as the use here of terminating representations of rational numbers in place of infinite recurring representations used in [5].
2. ALTERNATING SERIES REPRESENTATIONS FOR REAL NUMBERS.
For the convenience of the reader, because full previous details may be inaccessible to many (including the present authors), we prove here the fundamental results concerning the representation of real numbers via infinite alternating series.It is convenient to introduce here a more general alternating series, analogous to the positive series of Oppenheim [6], out of which we can deduce the results for alternating-Engel and alternating-Sylvester series as special cases.We define the alternating-Oppenheim algorithm as follows: Given any real number A, let a [A] A 1 A a o Then we recursively define where Herein [1 are positive numbers (usually integers).
The two cases of particular interest to us are those for which b 1, an, n > 1 (alternating-Engel series) and bn cn 1,n >___ i (the alternating- Sylvester series).Evy reaZ number A has a unique representation in the orm where =o+l-!+ l- a I a 2 a 3 > +i) al> i az+ a% Fu)uthermore every reag numbe A has a unique eprettion in the 6orm where PROOF.
Repeated application of the alternating-Oppenheim algorithm yields In particular by setting b n c n > 0 for i < n an+l i > an(a n + i) if Ar An+1 < 1 --0 as n --, since 1 and the sequence Furthermore An+1 a (a +1) al {an} is strictly increasing.It follows that A has an alternating-Sylvester which also may terminate.
Uniqueness of the representations follows from Proposition 2.3 on order below.{Various other interesting special cases of the alternating-Oppenheim algorithm will be treated in a separate article.) We deduce now an important result on the alternating series expansions for rational numbers.
PROPOSITION 2.2 The alternting-Sylvester and aternting-Engel sies tminate t a fbte numb o terms i an oy i A is rational.

PROOF.
Clearly any number represented by a finite expansion is rational.

Pia Pi
In the alternating-Sylvester case we now obtain r+1 eL qi/l aiqi Thus 0_<_--Pi+I qi-Piai < Pi Since {pi} is a strictly decreasing sequence of non-negative integers we must eventually reach a stage at which Pn+1 O, whence The result for the alternating-Engel series follows similarly from i+i qi-Piai qi+l qi We note that for rational numbers there is a possible ambiguity in the final term, analogous to that for continued fractions.We eliminate this as follows" CONVENTION 1.
We replace the finite sequence ((ao,a an)) by ((ao,a an_2,an_1+l)) in the case a n an_l(an_l+1).Similarly we replace (ao,a I ,a n by (ao,a I an_2,an_l+l) in the case a n an_l+1 Furthermore we identify A(I with its finite expansion (Uo,al a n) or ((ao,a i an)), respectively.
In order to be able to compare finite sequences of different lengths in size we introduce the symbol m with the following properties" For any r(I, r < co oo+ K to  ), or A ((ao,a I )) B ((bo,b I )).In both o/ these case, the coditon A<B l equivalent to" a2n < b2n or (i i) a2n+l > 2n+1 where i 2n or i 2n+1 is the fit index iO such that a.It now follows from A a o + a -l a --l + A'2n, B ao+ a --l l _ _ _ a 2 + B'2n >_--a.+ i, i > i, that A < B In the alternating-Engel case, from ai+I Note that if b2n m then B'2n 0 and the result remains valid in this case The result is proved in a similar fashion if (ii) holds.

CONSTRUCTIONS AND ORDER PROPERTIES
In the constructions below, standard facts about the ordered field Q of all rational numbers are taken as understood.With the results of Section 1 as initial motivation, we now define two sets E* and * and order relations on them as follows" Let E* be the set of all formal infinite sequences A (ao,al,a 2 of integers i such that ai+1 >_--a.+rfor i>_--1,a11.Also, let * be the set of all formal infinite sequences A ((ao,al,a 2 )) of integers such that al>_--1 and ai+l>--_ai(ai+l) for i 1.
Finite sequences (rational numbers) are included in our sets E* and x using Convention 2. We will frequently mae use of the property all sequences in E* and satisfy a. implies a.
for all j > i In both the sets E* and * we shall use corresponding lower-case letters to denote the "digits" of the elements of the respective sets, and we define A < B if and only if (i) a2n < b2n or (ii) a2n+l > b2n+l where i 2n or i 2n+1 is the first index >_--0 such that a. b.

PROOF.
We use the same argument in both cases.Firstly, trichotomy is obvious.Next let A < B and B < C. Suppose a r b r for and b r o r for r < ,b c (i) If i < j then a r c r for r < , and a < b c (i even) or a. > b. c. (i odd).
(ii) If i j then a r c r for r < j and a i < b i < c i (i even) or a > b z > ci ( odd).
(iii) If Z > j then a r c r for r < j, and a b < c ( even) or a.
b. > c. ( odd).Thus A < C in each case.
We may now introduce symbols <--_, > and , and define (east) upp bound and (9reotest) lome bound/, in the usual way.

PROOF.
First consider a non-empty subset X of E*, which is bounded above by a sequence B (bo,b I Assume Be. since otherwise there is nothing to prove.Now A <B for every AEX, and there is a largest index such that every AX with a b has a I b I a b.We may assume a b for some AX since otherwise (o,l,m,m is an upper bound for X, where o is the maximum value of for elements of X We now define c o b o c b If + 1 is odd let c+1 be the least possible value for the digit a+1 of any AX with a b o.If + 1 is even let c+1 be the greatest possible value for the digit +1 of any AX with a b o, where we take c+1 m if +1 has no largest value.In either case if c+1 m we are done, and put C (Co,C 1 c,m,m ).Otherwise we continue to define c+2 as the least possible value or greatest possible value depending on whether + 2 is odd or even, respectively, for the digit a+2 of all elements of the form (Co,C I C+l,a+2,a+3 in X. Again if c+2 m we are done and put C (Co,C 1 C+l,m,m ). Continue inductively, to define c++1 as the least possible value (++i odd) or.greatest possible value (+i+l even) for the digit a++1 of an element of X of the form (Co,C 1 c+Z,a++l,+Z+2 ).
If, when ++1 is even, a+i+1 has no largest value, we take c+Z+ m.The process terminates if at any stage c++1 m.We then take C (Co,C 1 c+,m,m ).Otherwise this process constructs a non-terminating The argument for S* is almost identical to the above, except that the sequence C ((Co,Cl,C 2 )) defined inductively via suitable elements of S* will now saisfy ci+ >_--ci(i+l),Cl >_--1.

EMBEDDING AND DENSITY OF RATIONALS
The following proposition justifies our use of Convention 1.The tntZng-EngZ and ntng-Sylvt ogoCtm dne 1 1 ord-prving maps PE* (I E* and PS* (I * whose images oe dense in E* and * respectively.

PROOF.
It is an immediate consequence of the results quoted earlier that the two algorithms define 1 1 maps pE (1--E* and PS* -* By Proposition 2.3 and the definition of order in E* and , these maps are then order-preserving.
Let k be the least index for which a k b k We show now in every possible case that we can find a rational number D satisfying A+B Now let A A < D < B. For A , B ( we take D- If k is even then k < We note that sequences in E* and x have the intuitively desirable property that rational numbers can be represented only by finite sequences (excluding m's) This des not hold in the case of ordinary Engel and Sylvester series (see for example [1] ).
APPROXIMATION LEMMA 4.2 Given anq eenent A o E* (rpecLve.y,*), there exist ationols A n or n >--_ 0 such (i) A (2m) A (2n) A A (2n+1) (ao,a I EE*, define the rational sequences A (n) by A (n) (ao,a I an,,m ).Then part(i) follows.Next suppose that A< B A (2n+1) for all n.In that case, we must have a > b if m is odd, or a < b if m is even, for the first index m such that a a (n>_-0) for any routional A.
For any A,B( E X (or A,B*) we now define A + B sup(A (2n) + B(2n)), A sup(-A(2n+l)), which exist in * (respectively, I*) because A (2n) + B (2n) A (1) + B (1) A(2n +1) < A (0) At this stage we note that the formal structures of the sets E* and x are very similar to the set K, (based on continued fractions) used by Rieger [4] to construct the real numbers.Thus to avoid repetition, we will refer the reader to the corresponding result of Rieger whenever the proof of the algebraic property there is the same.
LEMMA 5.1 The above operations make E (respectively, *) into an aelian group cow.raining ((I,+) as a dense subgroup.Fuuther Now we can represent finite sequences by infinite sequences as follows- m for j > n and CONVENTION an,m,m ).Similarly represent A ((a o,a I an)) (1 by A ((a o,a I an,, )).
,a1,a 2 )), and A n a n arian+1 anan+lan+2 as A (ao,al,a 2 ). (Note that we do not assume at this stage that A n A n dfined by either algorithm Now suppose (i) holds.If firstly a o < b o thenA a + A 1 < a + i b b + B 1 Bin either case.Next suppose a2n < b2n' n > O, in the alternating-Sylvester case.
we have CE* since Cr+l>--_c > a. (i even) i I c I i.Also if C A then C >A since either c or e. < a. (xL odd) for the first index >k such that c. a.(by the definition of the sequence Lastly, C sup)< since otherwise )< has an upper bound D< C. Then d r. c-,O < i < m,d m em If m is odd then dm > Cm Hence every element of the form A (eo,C Cm,am+l,am+2 in X satisfies D < A --<_ D contradiction.If m is even we have d < cIn the cases m 0 or e m < m (m > O) every element of the form A (e o,c1,....cm, m a m m for every A we can choose satisfies D < A <_--DIn the case e m For a > d we A (c e I, am_ am,am+ 1 )X with arbitrarily large a m.
Ol,...Ok,Ok+l+l,m,m if Ok+l m' or D (ao,a I ak,ak+l,k+2+l,m, m if Ok+l m' i.e., Bm{,A .If instead k is odd then ak > Ok; we choose D (=o,al ak,ak+l+l,m, m if ak+l m' or D (o,1 D,6k+1,O+2 +1, m,m if ai+l m and A{,B .A similar argument works in the case of that A (2n) =<C<A for all n Consider the first index m for which om# c m.If m is even we must > c which yields the contradiction A(m)<_--C < A (m) If m is odd we have a m m m+li have a m < Cm, which gives the contradiction A (m+l) <C < AThe same argument leads to parts (i) and (ii) for *For part (iii) in Ex, the formula for alternating-Engel series for rationals 1 >_--a/+l For *, the corresponding formula for alternating- Sylvester series of rationals gives instead A(2n+l)_ A(2n) ai+1 ->_-ai(ai+l)5.ALGEBRAIC OPERATIONS IN E* ANDSince we alreaady regard as an actual subset of * and * by Convention 1, it will simplify the discussion on algebraic operations below if we now re-deJrine A(n)