Additive basis of order $2$ (II)

Question

Can we find, using elementary ways, an additive basis of order $2$, $(u_n)_{n\geqslant1}$, such that $\lim\limits_{n\rightarrow+\infty}(u_{n+1}-u_n)=+\infty$ ?

If $\alpha\in\left]1,\frac32\right[$, the sequence $\left(\lfloor n^{\alpha}\rfloor\right)$ works (have a look here), but the proof is a little complicated for me...

Answer 1

I finally found this article : Cassels bases (cf. Theorem 6, p. 11).