Select $n^2 + 1$ points in the unit square. Show that at least two points are no more than a distance $\sqrt{2}/n$ apart

Question

I know I need to use the pigeonhole principle to prove this, but I don't know exactly how.

What I think I could do is divide the unit square into $n^2$ squares.

Using Pythagoras theorem, the maximum distance between 2 points in these squares is $\sqrt{2}/n$ (diagonals).

Then by the pigeonhole principle there must be a square with $2$ points in it since there are $n^2$ squares and $n^2 + 1$ points.

Would this be a correct way to prove the statement?

Answer 1

Your argument is correct.

Of course, you have to think about that the squares overlap, so a point could be in more than one square. But the pigeonhole principle still works if some pigeons get duplicated and put into multiple holes.