I am working through Turing computability by Soare, and have been stumped on an exercise asking you to show the weakness of the least function for a $\Delta_2$-approximation.
Let $A$ be a set, and $\{A_s\}_{s \in \omega}$ be the corresponding $\Delta_2$-approximation. The least function of $A$ and its approximation is denoted as $m_A(x)$. On input $x$, the least function is equal to the number of the first stage of the approximation which is correct on all numbers up to $x$. In other words,
$$m_A(x) = \mu s [\{ a: a \in A \wedge a \leq x \} = \{ a: a \in A_s \wedge a \leq x \}]$$
The exercise asks you to construct a non-computable $\Delta_2$ set $A$, a computable function $f$, and a $\Delta_2$-approximation $\{A_s\}_{s \in \omega}$ for $A$. You must show that $f(x)$ dominates $m_A(x)$.
I managed to create a $\Delta_2$-approximation for any $\Delta_2$ set $A$, by using the fact that $A \in \Sigma_2$ and $\bar{A} \in \Sigma_2$. However, creating $f$ to dominate $m_A$ escapes me.
I have considered an approximation $A_s$ which "brute forces" all possible combinations of elements belonging to it. For example, for each subset $N$ of $\{x \in \mathbb{N}: x \leq \log_2 s\}$, there exists a stage $i$ such that $A_i = N$. That is to say, the approximation generates a stage for each possible subset of $\omega$ up to a certain value. One of these stages must be correct up to $log_2 s$, as it includes all permutations. Here, the least function would be easy to dominate with a computable function. However, for such an approximation the modulus of convergence is not defined, as each element will always move back and forth between belonging to the set and not belonging. So this is a dead end too.
I am stuck on this question. Any help would be appreciated.