Number theory sequence proof

MachNugget

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Suppose A = (an) = (a1; a2; a3; : : : ) is an increasing sequence of positive integers.
A number c is called A-expressible if c is the alternating sum of a nite subsequence
of A. To form such a sum, choose a nite subset of the sequence A, list those numbers
in increasing order (no repetitions allowed), and combine them with alternating plus
and minus signs. We allow the trivial case of one-element subsequences, so that each
an is A-expressible.
Denition. Sequence A = (an) is an \alt-basis" if every positive integer is uniquely
A-expressible. That is, for every integer m > 0, there is exactly one way to express
m as an alternating sum of a nite subsequence of A.
Examples. Sequence B = (2^n-1) = (1; 2; 4; 8; 16; : : : ) is not an alt-basis because
some numbers are B-expressible in more than one way. For instance 3 = -1 + 4 =
1 - 2 + 4.
Sequence C = (3^n-1) = (1; 3; 9; 27; 81; : : : ) is not an alt-basis because some numbers
(like 4 and 5) are not C-expressible.
(a) Let D = (2^n -1) = (1; 3; 7; 15; 31; : : : ). Note that:
1 = 1, 2 = -1 + 3, 3 = 3, 4 = -3 + 7, 5 = 1 - 3 + 7,
6 = -1 + 7, 7 = 7, 8 = -7 + 15, 9 = 1 - 7 + 15, . . .
Prove that D is an alt-basis.
(b) Can some E = (2; 3; : : : ) be an alt-basis? That is, can you construct an alt-basis
E = (en) with e1 = 2 and e2 = 3 ?
(c) Can some F = (1; 4; : : : ) be an alt-basis? Justify your answer.
(d) Investigate some other examples. Is there some fair
 
I think that's "finite subsequence".

Failure to proofread, combined with failure to show work, doomed this question. A pity; it could have been interesting (though not easy to give mere hints for).
 
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