Start with filling the set from n+1 to 2n. There is no free number within the interval that you can choose cause it will divide the nunber that is twise as large, and you have only chosen n numbers.

Now, we can prove that for any set of n positive integer numbers <= 2n, where no of the numbers divides any other, and no more number can be added cause it will divide another of the numbers or be divided itself, it is impossible to change the value of one of the numbers making one place in the interval free to place yet another number in. And...

Or something like that, just continue on and I'm sure you will solve it