A Question in Pigeonhole Principle

Question

We have 6 points in a 3*4 rectangle.prove that at least two points exist which their distance is less than $\sqrt2$.

Answer 1

You can chose 6 points whose distance is at least $\sqrt 2$ apart.

Answer 2

For another counterexample, take the points $(0;0)$, $(4;0)$, $(4;3)$, $(0;3)$, $(1.1;\ 2)$ and $(2.9;\ 2)$ in Cartesian plane.