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I'm reading Baby Rudin, and I don't understand his proof of theorem 8.14. He starts by defining an auxiliary function and then asserts that $\frac{1}{2π}$ times the integral of the Dirichlect kernel on $[-π, π]$ is 1. Good so far. But then I don't know what that integral has to do with the rest of the proof. Furthermore, he finds an expression for $s_N(f; x)-f(x)$ and says that it comes from line 78 (picture included) but I don't know where it came from. What's going on?

start of Theorem and some relevant equations. Theorem is ar bottom of page

second page of Theorem, with offending equation

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He is using that $$ \frac1{2\pi}\int_{-\pi}^\pi f(x)\,D_N(t)\,dt =f(x). $$ Note the variables.

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    Ah! Because f(x) is a constant with regards to the integral – 2017-02-09
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    Exactly. A very common and useful trick. – 2017-02-09