real analysis - Bijection from $mathbb{N}$ to $mathbb{N}$ such that $f(n) geq n text{ for large } n$ or $f(n) leq n text{ for large } n$ holds.

Let us consider an arbitrary bijective map $f: \mathbb{N} \to \mathbb{N}$. Then which one of the following is correct?

$$\begin{align} & f(n) \geq n \text{ for large } n \\ & f(n) \leq n \text{ for large } n \end{align}$$ I know that $f(n)=n$ is a bijection. But if $f$ is any other bijection other than identity, which of the above must hold? It may be happen that both can hold for different $f$. Please give me some example or if any proof, of the above.

Answer

None! We can define $f$ in a way that it "swaps" every pair of numbers: $f(1)=2$, $f(2)=1$, $f(4)=3$, $f(3)=4$... This is a bijection, but you have $f(n)=n+1>n$ for $n$ odd whereas $f(n)=n-1 for $n$ even.

Hope this helps!