I am having a bit of trouble understanding the inductive summation step in the proof of Theorem 8 of Chapter 2 in Lang's Introduction to Diophantine Approximations (p. 31). In particular, Lang shows that for $N$ sufficiently large,
$$ \lambda(N) - \lambda(N - q) = \int_{N-q}^{N} \psi(t) dt + \theta_{N,q} \int_{N-q}^{N} \frac{\omega(t)^{1/2} g(t)^{1/2}}{t} dt, $$
where $|\theta_{N,q}| \leq c_1 + 5$ for some nonnegative constant $c_1$ coming from lemma 2. (See $\S3$ of chapter 2 of Lang's book for the notation.) Lang then posits that the argument can be repeated with $N - q$ instead of $N$, and so on inductively, until one finally gets
$$ \lambda(N) = \Psi(N) + \theta \int_{1}^{N} \frac{\omega(t)^{1/2} g(t)^{1/2}}{t} dt + O(1), $$
where $|\theta| < c_1 + 5$ and
$$ \Psi(N) = \int_{1}^{N} \psi(t) dt. $$
However, I am not sure I see how this inductive summation can work, because 1) it seems unclear that the procedure can be repeated for $N-q$ or later, smaller, integers in the procedure, since they need not be sufficiently large that the first part of the proof applies; and 2) it seems unclear how the second term in the ultimate equation for $\lambda(N)$ can be obtained, since certainly $\theta_{N,q}$ and $c_1$ do not remain the same at each step of the summation, e.g., it seems that $\theta_{N,q} \neq \theta_{N-q,q'}$ in general if $q'$ is chosen for $N-q$ like $q$ was chosen for $N$.
Does anyone know how to make sense of this step in the proof?