Let $f$ be a bijection $\mathbb{N} \rightarrow \mathbb{N}$ , prove there exist positive integers such that...

Question

Let $f:\mathbb{N} \rightarrow \mathbb{N} $ be a bijection.

Prove that there exist positive integers $a < a + d < a + 2d$ such that $f(a) < f(a + d) < f(a + 2d).$

Answer 1

Let a = $f^{-1}(1)$.

Let $b_1 = f^{-1}(2)$.

1) If $b_1> a$ set $d = b_1-a$. $f(b_1 + d) > 2$. Therefore $$a < b_1 = a + d < b_1+ d = a + 2d$$ and $f(a) < f(a + d) < f(a +2d)$.

2) If $b_1< a$, let $b_2= f^{-1}(3)$ and proceed as in 1) above. Repeat as needed until $b_m= f^{-1}(m+1)> a$

There are only finitely many natural numbers less than a, so for some $k$ $b_k$ must be greater than $a$. Setting $d = b_k-a$ gives us our solution.