Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with $$\lim \frac{a_{n+1}}{a_n}=2$$ such that $$P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}$$ has density $1$ for every cofinite subsequence $A'$ of $A$?

Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with $$\lim \frac{a_{n+1}}{a_n}=2$$ such that $$P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}$$ has density $1$ for every cofinite subsequence $A'$ of $A$?

This has been solved in the affirmative by ebarschkis in the comments (based on idea of Tao and van Doorn, also in the comments).

This was formalized in Lean by Barschkis using Aristotle.