Challenge

Write down the numbers 1,2,...,2n for some positive integer n. Now out of these choose n+1 numbers. Prove that out of these n+1 numbers there exists numbers a and b such that a divides b.

demonstration/example.
1,2,3,4  n=2 so n+1=3.
The only set that doesn't contain 1 is {2,3,4} and 2|4.

Consider 1, 2, ..., 2n for some n in the natural numbers.
These may be labeled 1, a(2), a(3), ..., a(n)
1|a(i) where 1 < i < n+1

Consider 2, 3, 4, ..., 2n for some n > 1 in the natural numbers.
The number of odds between 2 and 2n is (2n - 2)/2 = n - 1
Selecting n+1 numbers ensures that at least 2 of the selected numbers are even.
We'll call these evens e(1) and e(2), e(1) being the smaller.
There are two cases
1: e(2) = e(1)*2^k, thus e(1)|e(2)

2: e(2) != e(1)*2^k, so e(1) does not divide e(2)
at least one of these evens divides, or is divisible by an odd.