Proving Convergence of a limit related to Cross-Spectrum Estimation Proving asymptotic covariance of Cross-Spectrum Estimation I came across this mathematical problem:

Consider two constants $M,L$ such that  $M\to \infty, L \to \infty \hbox{ and } \frac{L}{M} \to 0$, And $\ell,s,r \in \mathbb{Z}$ $\nu_1,\nu_1 \in \mathbb{R}$

$$ \lim_{M\to +\infty} \left\{\frac{1}{L}\sum_{\ell=-M}^{M} U_M(r,\ell,s) w\left(\frac{\ell+s}{L}\right)\  w\left(\frac{\ell}{L}\right) e^{-2\pi i\ell(\nu_1-\nu_2)}\right\}$$

where  $r$, $s$ are constants and:

$$ U_M(r,\ell,s)= \left\{\begin{matrix}
0 &amp;  r&lt;-M+\ell+s\\
1-\frac{\ell+s+r}{M}&amp; -M+\ell+s\leq r \leq 0 \\ 
1-\frac{\ell+s}{M}&amp; 0 &lt; r \leq s\\ 
1-\frac{\ell+r}{M}&amp; s&lt;r \leq M-\ell\\
0 &amp;  r&gt;M-\ell 
\end{matrix}\right.$$
And $w(x)$ is a function which is even, bounded,  square-integrable and such that $w(0)=1$.

It has been proven [1] that this limit exists and it is equal to  0 for $\nu_1\neq\nu_2$ and :
$$=\int_{-\infty}^{\infty}w^2(x)dx $$
for $\nu_1=\nu_2$. 
I am trying to prove that replacing in the limit $U_M(r,\ell,s)$ by

$$V_M(r)= \left(1-\frac{|r|}{M}\right)$$

Produces the same result but I don't have the background to understand the mentioned proof, is there a simple way to prove this claim? I am only interested in the case of $V_M(r)$.

[1] Hannan, E. J. Multiple time series, vol. 38. John Wiley &amp; Sons, 2009. Proof of Theorem 9 chapter V, page 314.
A first thing to try is to reduce the unwanted variables. What happens if you set  $\ell=s=0$?

$$ U_M(r,0,0)= \left\{\begin{matrix}
0 &amp;  r&lt;-M\\
1-\frac{r}{M}&amp; -M\leq r \leq 0 \\ 
1-\frac{0}{M}&amp; 0 &lt; r \leq 0\\ 
1-\frac{r}{M}&amp; 0&lt;r \leq M\\
0 &amp;  r&gt;M 
\end{matrix}\right.$$

Luckily, this definition seems to fit with $V_M$, since the third inequality never happens. So your $V_M(r)$ is just a special case. Nothing more to do.