Trying to prove a sequence of functions is increasing.

Question

Put $\delta_n = 2^{-n}$.To each positive integer $n$ and each real number $t$ corresponds a unique integer $k = k_n(t)$ that satisfies $k\delta_n \le t < (k+1)\delta_n$. Define

$$\psi_n(t)= \begin{cases} k_n(t) \delta_n,& \text{if } 0 \le t < n\\ n, & \text{if} \ n \le t \le \infty \end{cases}$$

I want to prove that this sequence of functions is increasing, i.e., that $\psi_1 \le \psi_2 \le \psi_3 ...$ but the way $k_n(t)$ is defined is giving me some problems with a formal proof. Could I have a proof?

Answer 1

Firstly observe, by the definition of the integer part, that $k_n(t)=[2^nt]$ ($[\cdot]$ denotes the integer part).

Fix $n\ge 1$. We will show that $\psi_n(t)\le\psi_{n+1}(t)$ for every $t$.

$(i)$ $0\le t<n$: the inequality becomes $2[x]\le [2x]$, where $x=2^nt$; this is true by Hermite's identity: $[2x]=[x]+[x+\frac{1}{2}]$.

$(ii)$ $n\le t<n+1$: we have to show that $\frac{[2^nt]}{2^n}\le n+1$; this is obviously true (in $2^nt<2^n(n+1)$ we can pass to the integer part).

$(iii)$ $t\ge n+1$: the inequality $n\le n+1$ is obvious.