Unit square and the pigeonhole principle

[CupcakeFury]

Hello

I was wondering if someone was kind enough to help me understand this questions and solution because I feel like I am missing a couple of things.
The purpose of tackling this questions was recreational.

The problem states :
Given a unit square, show that if five points are placed anywhere inside
or on this square, then two of them must be at most sqrt(2)/2 units apart.

The Solution states to partition the square into 4 smaller 1/2 x 1/2 squares and by the pigeonhole principle, one of these smaller squares must contain at least two points.And since the diagonal of each
small square is sqrt(2)/2 that is the maximum distance between the two points.

But my question is what stops me from picking two points,out of those five, which are 0.9 units apart which is larger than sqrt(2)/2 ~0.7 units ?

Thank you

 

[stapel]

The problem states: Given a unit square, show that if five points are placed anywhere inside or on this square, then two of them must be at most sqrt(2)/2 units apart.

The Solution states: Partition the square into 4 smaller 1/2 x 1/2 squares. By the pigeonhole principle, one of these smaller squares must contain at least two points. And, since the diagonal of each small square is sqrt(2)/2, this is the maximum distance between the two points within that smaller square.

But my question is what stops me from picking two points,out of those five, which are 0.9 units apart which is larger than sqrt(2)/2 ~0.7 units?

You're welcome to pick whichever points you like. The "problem" doesn't ask you to prove that all points, or any pair of two points, must be within the given distance. It only asks you to prove that some pair of points must be within that distance.