Let $\ F(n)\ (\mbox{where}\ n\in\mathbb N:=\{1\ 2\ \ldots\})\ $ be the least cardinality $\ |A|\ $ of a set $\ A\subseteq\mathbb N $ such that:

$\ \min A=n $

$\ \sum_{x\in A}\frac 1x = 1 $

**QUESTION** Is set $\ \{n\in\mathbb N:\ \frac{F(n)}n\le 2\}\ $ finite?

Background:

Roughly speaking, $\ \frac {F(n)}n\ \ge\ e-1,\ $ where $\ e:=exp(1)\ $ is one of the Euler's constants, $\ 2.718\ldots.\ $ This induces a bunch of natural questions, I have restricted myself to just one. Please, feel free to enjoy addressing also related questions.