CIIM 2020 Problem 6

For a set $A$ , we define $A + A = \{a + b: a, b \in A \}$ . Determine whether there exists a set $A$ of positive integers such that $$\sum_{a \in A} \frac{1}{a} = +\infty \text{and} \lim_{n \rightarrow +\infty} \frac{|(A+A) \cap \{1,2,\cdots,n \}|}{n}=0.$$ NoteGoogle translated from http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales