I am going over the proof of the Dirichlet-Jordan criterion for pointwise convergence of Fourier series (in J. Duoandikoetxea's book Fourier Analysis). At a certain point he claims that $$ \int_0^{1/2}\big(\csc(\pi t)-\frac{1}{\pi t}\big)\,dt<+\infty $$ without further explanation.
One can verify this directly since the integrand has an elementary primitive, but it involves calculating the primitive of $\csc x$, which is tricky (at least the proof I once knew). Also, one could observe that $1/(\pi z)$ is the principal part of the Laurent series of $\csc z$ around $0$ and thus that the integrand is a holomorphic function in the unit disk, but that seems like an overkill.
Is there any elementary way of showing the above estimate? Am I missing some obvious argument here?
reuns showed one way in a comment, here is another: as $t\to 0$, $$\sin(\pi t) = \pi t+O(t^3)$$ hence
$$\frac{\sin(\pi t)}{\pi t} = 1+O(t^2)$$ and taking the reciprocal yields $$\frac{\pi t}{\sin(\pi t)} = 1+O(t^2)$$ Divide by $\pi t$ to get $$\frac{1}{\sin(\pi t)} = \frac{1}{\pi t}+O(t)$$ so the integrand tends to zero as $t\to 0$, in particular it is bounded.