I am attempting to prove the statement: "If $n$ is a positive integer, and you choose any subset of size $n+1$ from the set $\{1,2, \cdots, 2n\}$, it will always be possible to find two distinct elements in the subset such that one is a factor of the other".
I have found a method which works provided that the following proposition is true: if $a < b \leq n$ and $a$ does not divide $b$, then there exists some multiple of $a$ within the set $\{n+1,n+2,...,2n\}$ which is not also a multiple of $b$. How can I prove this proposition?