Interesting and challenging

n=126.
Partition the set as follows:

{1,2}
{3,4,5,6}
{7,8,9,...,14}
{15,16,...30}
{31,32,....62}
{63,64,.....,126}

If you want to take 7 elements from the above 6 sets,2 of them have to be taken from the same set.(Pigeonhole Principle)

Let these 2 elements be x and y with y≥x.

x/y has to be between 1 and 2 (as the sets have been constructed in that manner)