Kolmogorov complexity of prefixes of computable sequences Let the characteristic sequence of a set $A ⊆ \mathbb{Z^+}$ be the following infinite binary sequence:
$$χ_A = b_1b_2b_3\ldots,$$
whose $n$th bit is 1 if $n ∈ A$ And 0 otherwise.
Write $χ_{A,n}$ for the first $n$ bits of $χ_A$.
Prove for every decidable set $A \subseteq \mathbb{Z^+}$ there is a constant $c ∈ \mathbb{N}^+$ such that for all $n ∈ \mathbb{Z}^+$ the Kolmogorov complexity of $\chi_{A,n}$ satisfies
$$ C(χ_{A,n})≤\log n+c. $$

Since $A$ is decidable there is a lexicograhic enumerator. Given that, how would I construct a Turing machine $\pi_A$ satisfying $U(\pi_A)= χ_{A,n}$, where $U$ is the universal Turing machine, whose complexity is at most $\log n+c$?
Let $D$ be a decider for $A$, then in order to determine $\mathcal{X}_{A,n}$ it is enough to specify $n$ and an encoding of $D$. Given $D,n$, you can loop over all integers $i\in [n]$ and check whether or not $i\in A$ by simulating $D(i)$. Thus, the overall description of $\mathcal{X}_{A,n}$ consists of $n$, the encoding of $D$ and the encapsulating machine $M$ (described above) which uses $D,n$ to output $\mathcal{X}_{A,n}$. $n$ contributes at most $\log n$ bits to the description, and $D,M$ have some encoding of length $c$ independent of $n$, bounding the overall description length by $\log n + c$.