We have $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$ for all $(x,n) \in \mathbb{R} \times \mathbb{N}$. Establish $$\forall (x,n) \in \mathbb{R} \times \mathbb{N}, \quad \pi_n(x) \leqslant x < \pi_n(x) + \frac{1}{2^n}.$$
We have $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$ for all $(x,n) \in \mathbb{R} \times \mathbb{N}$.
Establish
$$\forall (x,n) \in \mathbb{R} \times \mathbb{N}, \quad \pi_n(x) \leqslant x < \pi_n(x) + \frac{1}{2^n}.$$