Discreteness and periodicity in Fourier transform Why discreteness in time / frequency domain dictates periodicity in the other frequency / time domian?


For example the DTFT is perodic in frequency? 
Why it doesn't contain all the frequencies? 
Why the DTFS contain only finite number of frequencies?


NOTE: 

I've read the derivation but sometimes i see it's like a definition or a claim in the case of DTFS.


How we prove that equation of DTFS ? 
$$
x[n] = \sum_{k=0}^{N-1} a_k e^{jl\omega_o n}
$$
This is, essentially, what the sampling theorem is about.  Uniform sampling in one domain (e.g. the time domain) causes periodic extension in the reciprocal domain (e.g. frequency domain).
The reason why is that the sampling function is a periodic function which means it can be represented as a Fourier series
$$\begin{align} 
\operatorname{III}_T(t) \ &amp;\triangleq\  \sum_{k=-\infty}^{\infty} \delta(t - k T) \\
 &amp;= \sum_{n=-\infty}^{+\infty} c_n e^{j 2 \pi n \frac{t}{T}}  \\
\end{align}$$
where the Fourier coefficients are
$$ \begin{align}
c_n\, &amp; = \frac{1}{T} \int_{t_0}^{t_0 + T} \operatorname{III}_T(t) e^{-j 2 \pi n \frac{t}{T}}\, \mathrm{d}t \quad ( -\infty &lt; t_0 &lt; +\infty ) \\[4pt]
&amp; = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \operatorname{III}_T(t) e^{-j 2 \pi n \frac{t}{T}}\, \mathrm{d}t \\[4pt]
&amp; = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \delta(t) e^{-j 2 \pi n \frac{t}{T}}\, \mathrm{d}t \\
&amp; = \frac{1}{T} e^{-j 2 \pi n \frac{0}{T}} \\[4pt]
&amp; = \frac{1}{T} \ .
\end{align} $$
Using the definition of Fourier Transform most common with electrical engineering
$$ X(f) \triangleq \int\limits_{-\infty}^{+\infty} x(t) e^{-j 2 \pi f t} \ \mathrm{d}t $$
$$ x(t) = \int\limits_{-\infty}^{+\infty} X(f) e^{+j 2 \pi f t} \ \mathrm{d}f $$
when sampling $x(t)$
$$\begin{align}
x_\text{s}(t) &amp;\triangleq x(t) \cdot T \cdot \operatorname{III}_T(t) \\
 &amp;= x(t) \cdot T \sum_{k=-\infty}^{\infty} \delta(t - k T) \\
 &amp;= T \sum_{k=-\infty}^{\infty} x(t) \ \delta(t - k T) \\
 &amp;= T \sum_{k=-\infty}^{\infty} x(kT) \ \delta(t - k T) \\
\end{align}$$
which shows how $x(t)$ is converted to samples $x(kT)$ and this is also true:
$$\begin{align}
x_\text{s}(t) &amp;\triangleq x(t) \cdot T \cdot \operatorname{III}_T(t) \\
 &amp;= x(t) \cdot T \cdot \sum_{n=-\infty}^{+\infty} \frac1T e^{j 2 \pi n \frac{t}{T}} \\
 &amp;= \sum_{n=-\infty}^{+\infty} x(t) \ e^{j 2 \pi n \frac{t}{T}} \\
\end{align}$$
the resulting Fourier transform is
$$ X_\text{s}(f) = \sum_{n=-\infty}^{+\infty} X\left(f - n\tfrac{1}{T} \right)
 $$
which is periodic in the frequency domain with period $\frac{1}{T}$.
Because of the Duality of the Fourier Transform, if can be also shown that sampling in the frequency domain causes periodicity in the time domain, which is essentially all that Fourier series is about.