Suppose the stated in [Jordan's lemma](Jordan's%20lemma.md) ,[1.](Jordan's%20lemma#^24cd25), [2.](Jordan's%20lemma#^1358c3), and [3.](Jordan's%20lemma#^3ea7e2), are true. Consider also the contour integral [along a circular path](Contour%20integrals.md#Integrating%20over%20a%20circular%20arc) with the integrand in the statement of the lemma, [$\lim_{R\rightarrow \infty} \int_{C_R}dz f(z)e^{iaz} = 0$](Jordan's%20lemma#^16a424) Plugging in the expression from condition 2. and setting the limits of the integral over the arc defined in condition 1., We obtain $\int_{C_R}dz f(z)e^{iaz}=\int_{0}^{\pi}d\phi f(Re^{i\phi})e^{iaRe^{i\phi}}iRe^{i\phi}$ ^f3014c We may then plug in [Euler's formula](Complex%20analysis%20(index).md#Euler's%20formula) in the exponential that's in the exponential so that $e^{iaRe^{i\phi}}=e^{iaR(\cos{\phi}+i\sin{\phi})}=e^{aR(i\cos{\phi}-\sin{\phi})}.$ If we then consider just the expression $e^{-aR\sin{\phi}}$ from [Jordan's inequality](Analysis%20(index).md#Jordan's%20inequality) we see that $e^{-aR\sin{\phi}}\leq e^{-2aR\frac{\phi}{\pi}}$ for $\phi$ in $\big[0,\frac{\pi}{2}\big]$. Thus $\int_0^{\frac{\pi}{2}} d\phi \, e^{-R\sin{\phi}} \leq \int_0^{\frac{\pi}{2}} d\phi\, e^{-2R\frac{\pi}{2}}= \frac{\pi}{2R}(1-e^{-R})\;\;\;\;\mbox{for}\;R>0$ yielding the following inequality, $\int_0^{\frac{\pi}{2}} d\phi \, e^{-R\sin{\phi}} \leq \frac{\pi}{2R}\;\;\;\;\mbox{for}\;R>0$ ^3fd493 Applying [condition 3.](Jordan's%20lemma#^3ea7e2). we can also see that $|e^{iaRe^{i\phi}}|\leq e^{-aR\sin{\phi}}$. And thus combining the three inequalities, $\Bigg|\int_{C_R}dz f(z)e^{iaz}\Bigg|\leq f_{max} R \int_0^{\pi}d\phi e^{-aR\sin{\phi}}<\frac{f_{max}\pi}{a}$ where we also considered the fact that for any [Riemann integral](Riemann%20integral.md), $|\int dx f(x)|\leq \int dx |f(x)|$. ^3fddcd Taking the limit as $R\rightarrow \infty$ in [condition 3.](Jordan's%20lemma#^3ea7e2).,we find $\lim_{R\rightarrow \infty}f_{max} R \int_0^{\pi}d\phi e^{-aR\sin{\phi}}=0$ and we see that here $f_{max}\rightarrow 0$ and thus, [$\lim_{R\rightarrow \infty} \int_{C_R}dz f(z)e^{iaz} = 0.$](Jordan's%20lemma#^16a424) $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\blacksquare$ ^de96e4 #MathematicalFoundations/Analysis/ComplexAnalysis/Functions #MathematicalFoundations/Analysis/ComplexAnalysis/Integrals #Proofs