I cannot understand some part of the period signal's Fourier transform.

Here this my note's methods,

> For periodic signal with period $T_0$, define as $s_{T_0}(t)$ as
$$ s_{T_0} (t) = 
\begin{cases}
s(t) for -T_0/2 <t<T_0/2\\
0 & \text{otherwise.}
\end{cases}
$$
$$s(t)=\sum_{n=-\infty}^{\infty}s_{T_0}(t-n T_0) = s_{T_0}(t) * \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$
>So the Fourier transform will be
>$$S(f)=S_{T_0}(f) \cdot f_0 \sum_{n=-\infty}^{\infty}\delta(f-n f_0) = f_0 \sum_{n=-\infty}^{\infty}S_{T_0}(n f_0) \delta(f-n f_0)$$
and the other methods is
$$x(t)= s(t) \cdot \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$
$$X(f)=S(f) * f_0 \sum_{n=-\infty}^{\infty}\delta(f- n f_0) = f_0 \sum_{n=-\infty}^{\infty}S(f-n f_0)$$

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I cannot understand why I have "$\cdot f_0$" after Fourier Transform and why "$\delta(f-n f_0)$" can transform to "$\delta(t-nT_0)$"? And the last questions is, for the periods signal, why the note can use Fourier transform directly with out considering the Fourier serials?