Largest Mutually Coprime Subset of Initial Segment of Natural Numbers

Theorem
Let $n \in \N$ be a natural number.

Consider the set $\N_n$ defined as:
 * $\N_n = \closedint 1 n = \set {1, 2, \ldots n}$

Let $Q_n$ be the largest subset of $\N_n$ such that no element of $Q_n$ is the divisor of another element of $Q_n$.

Let $\map f n$ be the cardinality of $Q_n$.

Then for sufficiently large $n$:
 * $0 \cdotp 6725 \ldots \le \dfrac {\map f n} n \le 0 \cdotp 6736 \ldots$