I am proofing some series convergence,but I am not sure the part of argument is valid or not.
Argument: I have a sequence of variables $\{b_{n}\}_{n=1}^{\infty}$ which $b_{n}\in \mathbb{Z}_{2}$. At 1st step, we have finite number of coefficients $a_{i1}\in \mathbb{Z}_{2}$ where at least one $a_{i}$ not equal to 0. Then we set up an equation $\sum_{i=1}^{m}a_{i}b_{i}=c_{1}\in \mathbb{Z}_{2}$ with $m$ is finite. Since number of variables is larger than number of equations, the dimension of the null space of this linear system is infinite.
At j-th step, $j\in \mathbb{N}$, we have $j$ different linear equation, i.e. $\sum_{i=1}^{m_{k}}a_{ik}b_{i}=c_{k}$. $m_{k}$ is strict increasing natural number with respect to index $k\leq j$. Now $j<m_{j}$,the number of variables are still more than number of equations. The dimension of the null space is infinite.
We can do this for all $j$, obtaining some $b_{n}$. Now I claimed that the dimension of null space is infinite, because at each step, there are always infinite number of variables left. If this is true, then the set of possible solution has cardinal $2^{\mathbb{N}}$, which is uncountable.
Do the argument(s) hold?