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

Each result cited is universally quantified over the data in its own statement. Let $K\in\mathbb{R}$ satisfy $\psi(z)\le K$ for every $z\in Z$.

Claim 1. Let $\beta\in\mathbb{R}$ be positive. The set $\{\Psi_{\beta}(z):z\in Z\}$ is nonempty because $Z$ is nonempty. For $z\in Z$ one has $0\le\beta D(z)$ by claim 5 of Elementary Arithmetic in an Ordered Field, hence

$$\Psi_{\beta}(z)=\psi(z)-\beta D(z)\le\psi(z)\le K ,$$

so the set is bounded above and its supremum $M(\beta)$ is a real number by the least upper bound property. Since $D(z_{0})=0$ we have $\beta D(z_{0})=0$ and $\Psi_{\beta}(z_{0})=\psi(z_{0})$, and a supremum is an upper bound of the set, so $\psi(z_{0})\le M(\beta)$.

Claim 2. Let $\beta,\beta'$ be positive with $\beta\le\beta'$ and let $z\in Z$. Then $0\le(\beta'-\beta)D(z)$ by claim 5 of Elementary Arithmetic in an Ordered Field, so

$$\Psi_{\beta'}(z)=\Psi_{\beta}(z)-(\beta'-\beta)D(z)\le\Psi_{\beta}(z)\le M(\beta).$$

Thus $M(\beta)$ is an upper bound of $\{\Psi_{\beta'}(z):z\in Z\}$, and since $M(\beta')$ is the least such bound, $M(\beta')\le M(\beta)$.

Claim 3. Let $\beta,\eta$ be positive and let $z\in Z$ satisfy $M(\beta)-\eta\le\Psi_{\beta}(z)$. Since $\beta D(z)-\tfrac{\beta}{2}D(z)=\tfrac{\beta}{2}D(z)$,

$$\Psi_{\beta/2}(z)=\psi(z)-\tfrac{\beta}{2}D(z)=\Psi_{\beta}(z)+\tfrac{\beta}{2}D(z)\ \ge\ M(\beta)-\eta+\tfrac{\beta}{2}D(z).$$

As $\tfrac{\beta}{2}$ is positive, claim 1 applies to it and $\Psi_{\beta/2}(z)\le M\bigl(\tfrac{\beta}{2}\bigr)$. Combining the two displays and rearranging gives

$$\tfrac{\beta}{2}D(z)\le M\bigl(\tfrac{\beta}{2}\bigr)-M(\beta)+\eta .$$

Claim 4. For $k\in\mathbb{N}$ put $b_{k}=M(\beta_{k})$ and $a_{k}=b_{k}-b_{k+1}$. Each $\beta_{k}$ is positive and $\beta_{k}\le\beta_{k+1}$, because $2^{k}\le2^{k+1}$ and $\beta_{0}$ is positive, so $0\le a_{k}$ for every $k$ by claim 2. By Elementary Properties of Series of Real Numbers (v) the partial sums of the series $\sum_{k=1}^{\infty}a_{k}$ are

$$s_{n}=b_{1}-b_{n+1}\qquad\text{for every }n\in\mathbb{N},$$

and by claim 1, $\psi(z_{0})\le b_{n+1}$, whence $s_{n}\le b_{1}-\psi(z_{0})$ for every $n$. The set of partial sums is therefore bounded above, so the series converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (i), and consequently the sequence $(a_{k})_{k\in\mathbb{N}}$ converges to $0$ by Elementary Properties of Series of Real Numbers (ii).