Here $\mathbb{N}=\{1,2,3,\dots\}$.
We say that a set $A\subset\mathbb{N}$ is an additive base of natural numbers if there is a positive integer $h\in \mathbb{N}$ such that every natural number can be written as $a_1+\dots+a_h$ for some (not necessarily distinct) $a_i\in A\cup\{0\}$.
Some famous examples of such additive bases are the $k$-powers (Waring's problem) and the set of primes including $1$ (a theorem by Schnirelmann).
All the examples I encountered so far had the same property. If $A\subset\mathbb{N}$ was an additive base and $a_n$ was the $n$-th term of the set then $$\lim_{n\to \infty}\frac{a_{n+1}}{a_n}= 1.$$
It wasn't very hard to prove that indeed this is true for every additive base. Then I started to look the reverse direction. I tried to find a set $A=\{a_n\}_{n=1}^{\infty}$ (the terms are in ascending order) with $a_1=1\in A$ and $\lim_{n\to \infty}\frac{a_{n+1}}{a_n}= 1$ which is not an additive base but I couldn't.
I would be glad if someone could enlightened me with such an example. Thanks in advance!