I'm working through Stein and Shakarchi's Fourier Analysis, and I was doing Exercise 20 from Chapter 3. [There is a little vague-ness as I believe the question is asking us to show that the maximum of the partial sums is equal to the given integral in the limit as $N \to \infty$.] Either way, this lead me to evaluate the following limit of an integral.
$$lim_{N \to \infty} \frac{1}{2} \int_0^{\pi/(N+1)} D_N(t)-1 dt$$
where $D_n(t)$ is the Dirichlet Kernel. We can evaluate the integral of 1 and it will clearly vanish, so now we must tackle the other integral.
The trick is to multiply on top and bottom by $t/t$ and absorb the 1/2 factor from out front. Namely, this is then:
$$\lim_{N \to \infty} \int_0^{\pi/(N+1)} \frac{sin((N+1/2)t)(t/2)}{tsin(t/2)}dt$$
Now it seems "obvious" that the justification is that in the limit the integral is in an epsilon interval about 0, so that we may approximate $\frac{t/2}{sin(t/2)} = 1$. However, this feels woefully unrigorous. Note that once this is done, w simply make a u-substitution, and the limit of the integral is $\int_0^\pi sin(u)/u du$. Therefore my question is how to make this argument rigorous.
Two notes on this are as follows. The first is that I request we not use the Dominated Convergence Theorem. This is because the progression of this series of books is such that this Theorem will only be introduced in Book 3, [and additionally, then we need to multiply by an indicator function so that $\int_0^{\pi/(N+1)} D_N(t)/2 dt = \int D_N(t)/2 \chi_{[0,\pi/N+1]}$, which we would still need to do more measure-theoretic arguments for (perhaps BCT or MCT)].
The second is that I have seen people discussing using the limit composition theorem, but I have not found a good reference for this in the multi-variable case. Namely, if we re-write our limit as:
$$\lim_{N \to \infty} I(\pi/(N+1), 0, f(Nt))$$
where $I$ is the integral, then I cannot find relevant conditions to ensure this converges. Please let me know your thoughts on it, and/or if there is some theorem to justify eliminating the ratio of $t/2 / sin(t/2)$!
Parts in [] are parenthetical.