If $A$ is a subset of the set of positive integers $\mathbb{N}$, there are (at least) two notions of what it means for $A$ to be *thin*:


1. $A$ is thin in the 1st sense if $\lim\sup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0$; and
2. $A$ is thin in the 2nd sense if $\sum\_{a\in A}\frac {1}{a} < \infty$.


As user Dieter Kadelka points out in the comments below, (1) does not imply (2). Does (2) imply (1)?


**EDIT.** Originally there was a 3rd notion of thinness ("$A$ is thin in the 3rd sense if $\prod\_{a\in A}\frac{a}{a+1} > 0$"), but David Loeffler pointed out that this is equivalent to notion 2 above (see comments).



Two notions of thinness of subsets of $\mathbb{N}$
Since $|A \cap \{1,\ldots,n\}| / n \leq \sum\_{a \in A} \frac{1}{a}$ 2. implies 1. But if $A$ is the set of primes 1. holds true and 2. is false.


