TITLE: Limit problem involving bijective function
QUESTION [5 upvotes]: Let $f: \Bbb N^{\star}\to \Bbb N^{\star}$ a bijective functions such that exists $$\lim _{n \to \infty} {\frac {f(n)} {n}}.$$ Find the value of this limit.
I noticed that, if $f(n)=n$, then the limit is $1$. I couldn't make more progress. Can you help me?

REPLY [3 votes]: Let $f$ be permutation of $\mathbb{N^*}$. Suppose the inequality $f(n) \leq n$ holds for only a finite set of integers $A$. Let $m = \max A$, then $f \left( [m + 1,\infty) \cap \mathbb{N} \right ) \subset [m + 2,\infty) \cap \mathbb{N}$, thus $f^{-1} (\left \{ 1,m+1 \right \}) \subset \left \{ 1,m \right \}$, which is not possible. 
Thus $f(n) \over {n}$ $ \leq 1$ holds infinitely many times.
Applying the same reasoning to $f^{-1}$ we get :  $\frac{f^{-1}(n)}{n} \leq 1 $ i.e $f(n) \over {n}$ $ \geq 1$ holds infinitely many times.
Since the limit exists, $\lim \frac{f(n)}{n} = 1$ .