I can´t solve this problem and I will appreciate any help.
Given a set with N (N >= 1) distinct natural numbers. In each step we increase the lowest number by 2 and for every other number X we add X+1 to the set. Then, we replace each pair X,X with X+1 until all numbers in the set are distinct.
Prove that after finite number of steps we end up with a set that has only a single number.
Example:
A = {1,2,5,6,7,10}
Increase the lowest number by 2.
A = {3,2,5,6,7,10}
For every other number X we add X+1 to the set.
A = {3,2,2+1,5,5+1,6,6+1,7,7+1,10,10+1}
A = {2,3,3,5,6,6,7,7,8,10,11}
We replace each pair X,X with X+1.
A = {2,4,5,7,8,8,10,11}
We again replace each pair X,X with X+1.
A = {2,4,5,7,9,10,11} (after first step)