I guess this is quite standard and probably easy for experts or young lovers of number theory.

For $A\subseteq\mathbb N$, denote by $d^+(A)$ its upper density, which is 
$$
d^+(A)=\lim\sup\_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}
$$
Now let $A=(a\_n)$ be an increasing sequence of natural numbers and let $d\_n$ be the $n$-th difference $d\_n=a\_{n}-a\_{n-1}$.

> 
> **Question:** Suppose that $d\_n$ is not bounded, is it true that $d^+(A)=0$? 
> 
> 
> 

More generally: if $d\_n$ is not bounded, is it true that any (additive) invariant mean on $\mathbb N$ takes value $0$ over $A$?

Thanks in advance,

Valerio