Dirichlet Integral/Proof 5

Proof
Let $M \in \R_{>0}$.

Define a real function $I_M : \R \to \R$ by:
 * $\ds \map {I_M} \alpha := \int_0^M \dfrac {\sin x} x e^{-\alpha x} \rd x$

Observe that for all $\alpha > 0$ and $M' \ge M > 0$:

By Cauchy's Convergence Criterion, the improper integral

exists.

On the other hand:

Thus:

Thus:

By $(1)$, for all $A > 0$:

exists.