Some extra details left out of the title:
Given a group $G$, a symmetric subset $A \subset G$ containing $1$ is called a $K$-approximate group if $|A^2| = |\{ab \mid a,b \in A\}| \leq K|A|$
We are additionally given a result of Mal'cev, which says:
For a soluble subgroup $G \leq GL_n(\mathbb C), \; \exists B \leq G, C \leq Upp_n(\mathbb C), x \in G$ such that $B = xCx^{-1}$ and $[G:B] = O_n(1)$
Finally, our set up is that we have a finite $K$-approximate group $A \subset GL_n(\mathbb C)$ that generates a soluble subgroup of $GL_n(\mathbb C)$. I am asked to use Mal'cev's result to show that there exists a nilpotent subgroup $N \leq \langle A \rangle$ such that $N$ has step at most $n$ and $A$ is contained in the union of at most $K^{O_n(1)}$ left-cosets of $N$.
Now, since we know $G = \langle A \rangle$ is soluble, this immediately makes me think we are supposed to use Mal'cev here, giving us $B \leq G, C \leq Upp_n(\mathbb C), x \in G$ such that $B = xCx^{-1}.$
Now focusing on $C$, I know that if it is finite, then a Theorem by Breuillard-Green tells us that there exists a nilpotent subgroup $N_1 \leq Upp_n(\mathbb C)$ of step $\leq n$ such that:
$$|C^{O_n(1)} \cap N_1| \geq K^{-O_n(1)}|C|$$
This seems close to what we want as we have the restriction that $N_1$ has step at most $n$. Additionally, $C, N_1$ being groups means that $N_2 = C \cap N_1$ is a nilpotent group, and thus $N = xN_2 x^{-1}$ is a nilpotent subgroup of $B \leq G$.
This way we at least see that we can find a nilpotent subgroup of $G$ of step at most $n$. Regarding the condition about left-cosets, I am not too sure how I would go about showing this. Additionally, I don't really know what to do in the case that $C$ is infinite. I know this is equivalent to the case that $\langle A \rangle$ is infinite, as $B$ has finite index inside $\langle A \rangle$. However, I suspect that it is not true that the subgroup generated by a $K$-approximate group is always finite, as we can see from $A = \{-1, 0 , 1\} \subset \mathbb Z$, which will generate $\mathbb Z$.
I would really appreciate any help that may be offered to nudge me in the right direction, thank you.