$\DeclareMathOperator{\sinc}{sinc}\DeclareMathOperator{\asinc}{asinc}$I'm trying to follow the proof of the Fourier Transform for an Impulse train given in Julius Smith's textbook. I come across the following:
\begin{align}(2M+1)\asinc_{2M+1}(2\pi f) &= \frac{\sin [ \pi f(2M+ 1) ]}{\sin(\pi f)} \\&= \sum_{k=-\infty}^{\infty} \sinc(2Mf - k) \tag{1} \\&\rightarrow \sum_{k=-\infty}^{\infty} \delta (f -k) \tag{2}\end{align}where I suppose $\rightarrow$ in (2) denotes the limit as $M \rightarrow \infty$, and $\asinc$ denotes the aliased sinc.I'm not able to understand both of (1) and (2). In particular, it seems to me that$$\lim_{M \rightarrow \infty} \sinc (2Mf - k) = \lim_{M \rightarrow \infty} \delta(f - k/2M) = \delta(f). $$
Can someone help me?