SCHWEITZER 1972 Problem 3

Let $\lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form $$\mu= \sum_{i=1}^{\infty}n_i\lambda_i ,$$ where $n_i \geq 0$ are integers and all but finitely many $n_i$ are $0$ . Let $$L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ M(x)= \sum _{\mu \leq x} 1 \ .$$ (In the latter sum, each $\mu$ occurs as many times as its number of representations in the above form.) Prove that if $$\lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,$$ then $$\lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.$$

G. Halasz