We've had a class today about decomposing a real number in base $b$ and saw the following theorem :
Let $b\in\mathbb{N},b\geq2$.
For all $x\in\mathbb{R}_+$, there exists a unique sequence $a\in\mathbb{R}^\mathbb{N}$ such as:
- $\forall k\in\mathbb{N}^*,a_k\in[|0,b-1|]$
- $\forall n\in\mathbb{N},\sum\limits_{k=0}^n\frac{a_k}{b^k}\leq x\leq\sum\limits_{k=0}^n\frac{a_k}{b^k}+\frac{1}{b^n}$
- $a$ isn't stationary to $b-1$ (ie there is no $n_0$ such that $a_{k\geq n_0}=b-1$)
$(a_k)_\mathbb{N}$ is called the development of $x$ in the base $b$
$x=\overline{a_0,a_1a_2\cdots a_n}$ is the b-adic decomposition of $x$.
The first two points can be proved quite easily by building the sequence from scratch
$a_0=\lfloor x\rfloor$ in base $b$ and $a_{n+1}=\lfloor b^{n+1}\left(x-\sum\limits_{k=0}^n\frac{a_k}{b^k}\right)\rfloor$.
However, how can the 3rd point be proven easily ?
We went through a really really long proof in class, and i feel like there is an easier way.
Thank you.