Pointwise eventual chronological entry is uniform on compact subsets

Use the whole real-indexed family $(A_t{)}_{t\in \mathbb{R}}$. For each $p\in L$, the pointwise eventual-entry hypothesis supplies a real threshold $a$ such that $p\in A_t$ for every $t\ge a$; in particular $p\in A_{a+1}$. Hence $L\subseteq {\bigcup }_{t\in \mathbb{R}}{A}_t$. Each member is ambient open by [F1], so this is an indexed ambient open cover as in [F2]. If $L$ is nonempty, the indexed form of [F2] gives finitely many real indices $t_1,\dots ,t_k$, with $k\ge 1$, such that $L\subseteq A_{t_1}\cup \cdots \cup A_{t_k}$.

If $L$ is empty take $t_L=0$. Otherwise the finitely many real indices in step 1.1 have a greatest member: start with $t_1$ and compare each subsequent index with the retained member, retaining the larger one at each of the finitely many comparisons. Call the resulting member $t_L$; then $t_L\ge t_i$ for all $i$. For every $t\ge t_L$, monotonicity gives $A_{t_i}\subseteq A_t$ when $t_i<t$, and equality handles $t_i=t$. The finite cover therefore gives $L\subseteq A_t$. This proves the asserted compact uniformity without strengthening the pointwise hypothesis or using integer cofinality.