Let points be p1,p2,....,p8

placing point p8 at the centre of the circle and placing other seven points on the periphery of the circle, at equal distances.

Now we have one point at the centre seven points at the periphery of the circle. The heptagon which will be formed by joining the seven points on the circle will have all its side equal in length. The length of the side of the heptagon is less than 2pi/7 cm which is less than 1 cm.

Now if we replace the position of any point then too we get a pair of points whose distance from each other is less than 1 cm.