*elementary methods in analytic number theory*, 1965 (translated by Amiel Fienstein/Revised and edited by L. J. Mordell).

**Definition 5**:

*Additive*

*Basis*

A subset $A \subseteq \mathbb{N}$ with the property that $A + A + \cdots + A = \mathbb{N}$ for a finite sum.

**Theorem 1.1.2**(

*L. G. Shnirelman*)

*Every sequence with a positive density is a basis for the natural numbers.*

Proof:

(1) Let $A^{(k)} = \underbrace{A + A + \dots + A}_{k\,\text{times}}$

(2) Using the Corollary from Theorem 1.1.1, we have:

$$d(A^{(k)}) \ge 1 - (1 - d(A))^k$$

(3) For sufficiently large $k$, we have:

$$d(A^{(k)})>\frac{1}{2}$$

since $0 \le\frac{A(n)}{n}\le 1$ and $d(A)=\inf\limits_{n}\frac{A(N)}{n}$.

(4) For this $k$, the number of terms of $A^{(k)}$ in the interval $[1,n]$ will be greater than $\frac{n}{2}$.

(5) If $a_i$ ranges over the terms of $A^{(k)}$ which do not exceed $n$, then there exists $i_1,i_2$ such that: $a_{i_1} = n - a_{i_2}$.

Note: If $i_1,i_2$ don't exist, then there are more than $2*(\frac{n}{2}+1)=n+2$ values in $[0,n]$ which is impossible.

(6) So, there exists $k$ such that for any $n$, there exists $a_{i_1}, a_{i_2} \in A^{(k)}$ such that $n = a_{i_1} + a_{i_2}$.

QED

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