Is there any problem related to this? Or advice on how to approach a problem like this.
Given a set $A$ that contains the series $\{a_n\}_{n\in\mathbb N}$, prove whether it is possible to write the elements of another set $B$ as a sum of $k$ combinations of $a_n$.
For example, I have an arbitrary series $\{a_n\}$, how do I prove/disprove that the elements of $B$ can be written as: $$b_n=\overbrace{a_1+a_2+a_3+...}^{k \text{ times}}$$
Specifically, in my case, set $B$ contains all natural numbers, how do I approach the problem? i.e. to show that: $$\begin{align} 1=&\text{combination of $a_n$}\\ 2=&\text{combination of $a_n$}\\ &\vdots \end{align}$$