I am trying to understand the proof the following statement:
Let $p\ge2$ be a fixed natural number. Then, every natural number $a$ can be uniquely represented in the form: $$a=c_np^n+c_{n-1}p^{n-1}+\dots+c_1p+c_0,$$ where $c_n>0, 0\le c_i<p,\forall i=0,1,\dots,n$.
There's something I cannot understand at all:
Uniqueness:
Let's suppose we have two distinct representations: $$a=c_np^n+\dots+c_0,c_n>0,0\le c_i<p$$
$$a=b_np^n+\dots+b_0,b_n>0,0\le b_i<p$$
$\Rightarrow p | b_0-c_0 \Rightarrow b_0=c_0$
$$\Rightarrow c_np^n + \ldots + c_1 = b_np^n + \ldots + b_1 \Rightarrow p | b_1 - c_1 \Rightarrow b_1 = c_1 \text{ and so on.} $$
I don't understand what's going on in the proof. How do we conclude that $p$ divides $b_0-c_0$ and how does this mean that $b_0=c_0$? What do we do next?
