Covering compact subsets of plane with certain rectangles Suppose $\{F_n\}$ is sequence of compact subsets of the half-plane $\mathbb H=(0,+\infty)\times(-\infty,+\infty)$ with positive area such that $F_n\subset F_{n+1}$ and $\cup_n F_n=\mathbb H$. Do there exist a sequence of positive numbers $\{a_n\}$ and positive constants $\delta,\delta',\delta''$ such that 
$$[\frac{1}{a_n},a_n]\times[0,a_n]\subset F_n\subset[\frac{\delta}{a_n},\delta'a_n]\times[-\delta'' a_n,\delta''a_n]$$
for $n>n_0$?

No. Let $c_n$ be a monotone increasing sequence. For convenience let $d_n = \frac{c_n}{c_n-1}$. Then set
$$F_n = \bigg(  [c_n^{-1},c_n]\times[-c_n, c_n] -  [d_n^{-1},d_n]\times[-d_n, d_n] \bigg) \cup \{1\}\times [-c_n, c_n] $$
Each $F_n$ is closed, bounded, so compact. It looks like a large rectangle with a smaller rectangle punched out and a line strung across the removed smaller rectangle. (The line there means that the $F_n$ indeed converge to $\Bbb{H}$.) As $n$ increases, the large rectangle lets larger and the removed rectangle gets smaller. 
Because the inner rectangle you propose has $(1,\varepsilon)$ in its interior for small $\varepsilon$, whereas $(1,\varepsilon)$ is on the boundary of $F_n$ for each $n$, containment can never hold.
