Let $f:\mathbb{N}\to\mathbb{Z}$ be a bijection.
Proposition $1$: For any $m>0,\ \exists\ $ infinitely many $N', N$ with $N'>N,\ $ such that $ f(N') - f(N) \geq m(N'-N).$
Proof: Eventually there will exist a block of length $\ m:\ a,a+1,\ldots,a+m-1,\ $ such that all these are in $\{ f(u): u\leq q\}\ $ for some $q\in\mathbb{N}.$ Then, there must exist $v>q$ s.t. $f(v) < a$ and $f(v+1) > a+m-1,\ $ and so we are done: $N=v,\ N' = v+1.$
Proposition $2$: $\ \exists\ $ infinitely many $N', N$ with $N'>N,\ $ s.t. $ f(N') - f(N) \geq N(N'-N).$
Counter-example: $f(1)=0,\ f(2) = -2,\ f(3)= -1,\ f(4)= 1,\ f(5)= -4,\ f(6)= -3,\ f(7)= 2,\ f(8)= -6,\ f(9)= -5,\ f(10)= 3,\ f(11)= -8,\ f(12)= -7,\ f(13)= 4,\ $ and so on. In fact, in this example, there are no $N,$ let alone infinitely many.
Proposition $3$: $\ \exists\ $ infinitely many $N', N$ with $N'>N,\ $ s.t. $ \vert f(N') - f(N)\vert \geq N(N'-N).$
Proof/counter-example: ??
