Jabarkan Kernel Dirichlet dari Integral Dirichlet

Jelasnya,  adalah kontinu jika , untuk melihat keberlanjutannya di 0 terapkan Aturan L'Hopital:

Karenanya,  memenuhi persyaratan Riemann-Lebesgue Lemma. Ini berarti

(Bentuk Lemma Riemann-Lebesgue yang digunakan di sini dibuktikan dalam artikel yang dikutip.)

Pilih batasan  and . Maka kami ingin mengatakan

In order to do so, however, we must justify switching the real limit in  to the integral limit in .  This is in fact justified if we can show the limit does exist, which we do now.

Using integration by parts, we have:

Now, as  and  the term on the left converges with no problem. See the list of limits of trigonometric functions.  We now show that  is absolutely integrable, which implies that the limit exists.

First, we seek to bound the integral near the origin. Using the Taylor-series expansion of the cosine about zero,