Prove by induction that $$\forall n \in \mathbb{Z}^+, \forall x \in [0, 1), 0 \leq x - \frac{\lceil 2^n x \rceil - 1}{2^n} \leq \frac{1}{2^n}.$$
I have to prove two things I believe, that $0 \leq x - \frac{\lceil 2^n x \rceil - 1}{2^n}$ and $x - \frac{\lceil 2^n x \rceil - 1}{2^n} \leq \frac{1}{2^n}$, but the ceiling function is really throwing me off. Can anyone suggest any tips to go about this?
$y\leqslant \lceil y\rceil\leqslant y+1$ for all $y\in\mathbb{R}$ thus $2^nx\leqslant\lceil 2^nx\rceil\leqslant2^nx+1 $ and $$ 0\leqslant x-\frac{\lceil2^n x\rceil-1}{2^n}\leqslant\frac{1}{2^n} $$