Find the limit of $\dfrac{\phi(n)}{n}$ where $\phi$ is a permutation of $\Bbb N$

Question

Find the limit of $\dfrac{\phi(n)}{n}$ where $\phi$ is a permutation of $\Bbb N$.
A permutation of $\Bbb N$ is a bijection from $\Bbb N$ to $\Bbb N$, based on pure intuition it seems like the limit should be $1$ however I can't come up with a proof.
An already established result is that $(\phi(n))_n$ diverges to infinity.

Answer 1

Here is one without limit: Partition the natural numbers at the powers of 2. Let the permutation reverse each partition. The fraction $\frac{\phi(n)}n$ will bounce between $2$ and $\frac12$ infinitely many times.

If you pick partitions that grow in size faster than exponential (say you partition at each factorial), then the fraction goes arbitrarily close to $0$ and arbitrarily large as $n$ grows.