Penalised Suprema: Monotonicity, Near-Maximisers, and Vanishing Penalty along a Doubling Sequence

In the setting of The Real Numbers: Standing Notation and Background, let $Z$ be a nonempty set, let $\psi:Z\to\mathbb{R}$ be bounded above, let $D:Z\to\mathbb{R}$ satisfy $0\le D(z)$ for every $z\in Z$, and suppose there is $z_{0}\in Z$ with $D(z_{0})=0$. For a positive $\beta\in\mathbb{R}$ let $\Psi_{\beta}:Z\to\mathbb{R}$ be the function with value

at $z\in Z$. Then the following hold, where $M(\beta)$ denotes the supremum of the set $\{\Psi_{\beta}(z):z\in Z\}$ and $2^{k}$ is the natural power.

1. (The suprema are defined)¶ For every positive $\beta\in\mathbb{R}$ the set $\{\Psi_{\beta}(z):z\in Z\}$ is nonempty and bounded above, so that $M(\beta)$ is a real number, and $\psi(z_{0})\le M(\beta)$.

2. (Monotonicity)¶ For all positive $\beta,\beta'\in\mathbb{R}$ with $\beta\le\beta'$ one has $M(\beta')\le M(\beta)$.

3. (Near-maximisers have small penalty)¶ Let $\beta,\eta\in\mathbb{R}$ be positive and let $z\in Z$ satisfy $M(\beta)-\eta\le\Psi_{\beta}(z)$. Then

4. (Vanishing along a doubling sequence)¶ Let $\beta_{0}\in\mathbb{R}$ be positive and let $\beta_{k}=2^{k}\beta_{0}$ for $k\in\mathbb{N}$. Then the sequence whose $k$-th term is $M(\beta_{k})-M(\beta_{k+1})$ converges to $0$.