derivation of discrete Fourier Transform
The Discrete Fourier Transform is one-and-the-same as the Discrete Fourier Series.  The DFT (and explicitly DFS) maps one periodic sequence, $x[n]$ of period $N$:

$$ x[n+N] = x[n] \qquad \forall n \in \mathbb{Z} $$

to another periodic sequence, $X[k]$, having the same period $N$:

$$ X[k+N] = X[k] \qquad \forall k \in \mathbb{Z} $$

and the inverse DFT (or "iDFT") maps it back.

I have a pretty rigid Nazi-like position about it (see the other answer), but I don't think that the math cares.  Now if you wanna connect this to the more common (but often misinterpreted) derivation, you can look at a previous answer of mine.

The term "support of $x[n]$" simply means the domain of values of $n$ in which $x[n]$ (presumed to be non-periodic and to have "finite support") is potentially non-zero.  It could also mean the duration of $x[n]$.  But when you periodically extend $x[n]$, then that sequence has no finite support.  $x[n]$ is defined and potentially non-zero everywhere (or all integers).