The word "noice" is puzzling. Can you check it is not a typo for either "nice" or "noise" or even something else?

Sure, I have done the correction, it was a typo haha. Thanks.

$<z_i>$ is a finite alternating subsequence of $<z_1,z_2,z_3,\ldots>$ and its sum is $z_i$

@Permutator: What is the source of this problem?

I'm not sure about the source coz I found it on one of my discord server. Thanks

I think we need to characterize the sequence here. I mean characterizing some terms in the sequence like showing that if it is a nice sequence then it has infinitely many terms of the form XYZ.

No, list should be in increasing order not decreasing.

We allow the trivial case of one-element subsequences.

How do you specify alternating sequence. For instance is $z_1-z_2$ possible or should the minus sign arise only when there are $3$ terms as in $z_1+z_2-z_3$. In other words do we have $\pm z_1\mp z_2\pm z_3\mp\cdots$ or $z_1+z_2-z_3+z_4-z_5+\cdots$ or $z_1-z_2+z_3-z_4+z_5-\cdots$ ?

@zwim Alternating sequence is specified to be: $\pm z_1 \mp z_2 \pm z_3 \mp z_4 $. So like if $z_1$ is negative then $z_2$ should be positive and then $z_3$ should be negative and son on.. Thanks!

Your comments are contradictory; do you allow one-element subsequences or not?

Yes I do. Thanks.

If you allow one-element subsequences, isn't then trivially every $\mathcal{D}$ nice as miracle173's comment points out?

Ok then it's not allowed

@Vepir I don't think that a comment from miracle173 implies that every sequence is nice. Finger pointing (on my part, to a particular sequence) is certainly not nice at all, but you cannot get an odd number starting with the sequence of even numbers.

@Mirko Why would you need to get an odd number if your sequence is of even numbers? The question requires to represent all integers in $\mathcal{D}$ using alternating sums from $\mathcal{D}$.

@Permutator Your last edit says that one element subsequences are allowed, but your last comment says the opposite. Which one is it then? Also, do you care about representing only integers in $\mathcal D$ or is "$z_i \in \mathcal{D}$" a typo and you want to represent all positive integers $z_i \in\mathbb N$?

@Vepir If only integers in $\mathcal{D}$ need to be represented (which I had overlooked) then indeed one should NOT allow one element subsequences (if the problem is to be non-trivial). The sequence $1,2,4,8,16,...,2^{n-1},...$ is interesting in that it could represent every integer, and in particular its own elements, $2^{n-1}=-2^{n-1}+2^n$ (whether or not you allow one element sequences.) $3=-1+4, 5=1-4+8, 6=-2+8, 7=-1+8, 9=16-7$ and plug in the representation for $7$ and distribute the $-$ sign, $9=16-(-1+8)=1-8+16$, etc.

@quasi Seueqences with $z_{i+1}-z_i=1$ infintiely often are certainly not nice - note that the OP's definittion of "nice" includes uniqueness!

@Hagen von Eitzen: Yes I missed the "uniqueness" condition. I'll delete that comment. Thanks.

Alright I can see the confusion, we need to represent all positive integers uniquely using the alternating sum of a finite subsequence of D

Sorry everyone, my bad I came to know about it right now. Please do understand

I have now edited the question. Thanks.

@Vepir so in the last edition of the problem, we allow one-element subsequences, but the catch is that we insist on uniqueness of the representation, and the answer by Hagen von Eitzen is that there are no such nice sequences. And, also, we need to represent every positive integer, not only those in $\mathcal D$. (And I am so confused at this point, so I need to read it all over, to see what the question asks, and whether it was indeed answered, and which sequence is an example of what.)

@Mirko Allowing one-element subsequences but requiring all integers to be representable and requiring uniqueness, then becomes a duplicate of Aternating sum of an increasing sequence of positive integers ? - and that answer is incorrect as of the last edit because then $2^n-1$ is a nice sequence example as shown in Alternating sum of positive integers.