$a_k$ is a countable sequence of positive numbers st. $\gcd(a_1,a_2,...)=1$, show exists a finite subsequence $a_{i_1},a_{i_2},...,a_{i_n}$ with gcd 1 The question is,

Let $a_k$ be a countably infinite sequence of positive numbers st. $\gcd(a_1,a_2,...)=1$, does it exist a finite subsequence $a_{i_1},a_{i_2},...,a_{i_n}$ with $\gcd(a_{i_1},a_{i_2},...,a_{i_n})=1$?

or more generally,

Let $a_k$ be a countably infinite sequence of positive numbers st. $\gcd(a_1,a_2,...)=d$, does it exist a finite subsequence $a_{i_1},a_{i_2},...,a_{i_n}$ with $\gcd(a_{i_1},a_{i_2},...,a_{i_n})=d$?

Can anyone help with an elementary proof (pls. do not use advanced number theory stuff) or a counter example? Thank you.

Hint: Let $d_n=\gcd(a_1,\dots,a_n)$. Then since $\{d_n\}$ is a non-empty set of positive integers, it must have a least element.
