Can anyone verify whether the following proof is correct or incorrect? If incorrect please explain where and why I went wrong?
Use mathematical induction to show that given a set of n+1 positive integers, none exceeding 2n, there is at least one integer in this set that divides another integer in the set.
S = {
such that i = 1,2,...,n+1,
2n}
Show that
such that

i
j
Basis Step: P(1): case 1: S={1,2} 1  2 so P(1) is true
case 2: S={1,1} 1  1 so P(1) is true
Inductive Step: P(k)
P(k+1)
Given S = {
such that i = 1,2,...,k+1,
2k}
such that

i
j
Prove that
such that

, s
t
where
{
such that i = 1,2,...,k+1,(k+1)+1
2(k+1)}
Note that { such that i = 1,2,...,k+1,(k+1)+1 2(k+1)} = S { such that i = (k+1)+1 2(k+1)}
Choose = that divides and = from the given part.
This proves P(k+1) is true, so the original statement is true by M.I.