[proofplan] Rapid decay means that every power $h^N$ bounds the $X$-norm of $u_h$ after possibly shrinking the range of $h$. The uniform boundedness hypothesis lets us pass this bound through $A_h$ with one constant independent of $h$. Since the exponent $N$ is arbitrary, the resulting $Y$-norm estimate is exactly rapid decay for $A_hu_h$. [/proofplan] [step:Fix an arbitrary decay order and record the $X$-norm estimate] Let $N\in\mathbb{N}$ be fixed. Since $u_h=O(h^\infty)$ in $X$, there exist constants $C_N>0$ and $h_N\in(0,h_0]$ such that \begin{align*} \|u_h\|_X \leq C_N h^N \end{align*} for every $h\in(0,h_N]$. [guided] We prove rapid decay in $Y$ by checking one decay order at a time. Fix an arbitrary $N\in\mathbb{N}$. The definition of $u_h=O(h^\infty)$ in $X$ says precisely that this chosen power $h^N$ controls the norm of $u_h$ once $h$ is sufficiently small. Therefore there are constants $C_N>0$ and $h_N\in(0,h_0]$ such that \begin{align*} \|u_h\|_X \leq C_N h^N \end{align*} for every $h\in(0,h_N]$. The constants may depend on $N$, which is allowed in the definition of rapid decay. What matters is that after $N$ is fixed, the estimate holds uniformly for all sufficiently small $h$. [/guided] [/step] [step:Apply the uniform operator bound on the common small-parameter range] Define \begin{align*} \widetilde h_N := \min\{h_N,h_1\}. \end{align*} Then $\widetilde h_N\in(0,h_0]$. For every $h\in(0,\widetilde h_N]$, both the rapid decay estimate for $u_h$ and the uniform operator estimate for $A_h$ apply. Hence \begin{align*} \|A_hu_h\|_Y \leq C\|u_h\|_X. \end{align*} Using the estimate from the previous step gives \begin{align*} \|A_hu_h\|_Y \leq C C_N h^N \end{align*} for every $h\in(0,\widetilde h_N]$. [/step] [step:Conclude rapid decay in $Y$ because the decay order was arbitrary] For the fixed $N\in\mathbb{N}$, set \begin{align*} D_N := C C_N. \end{align*} The previous step proves that \begin{align*} \|A_hu_h\|_Y \leq D_N h^N \end{align*} for every $h\in(0,\widetilde h_N]$. Since $N\in\mathbb{N}$ was arbitrary, this is exactly the definition of $A_hu_h=O(h^\infty)$ in $Y$. [/step]