The projective geometry method Without loss of generality we can view the desert as a plane surface. We project the surface onto a line and afterwards the line onto an interiour point of the cage. Thereby the lion is mapped onto that same point. The Bolzano-Weierstrass method Divide the desert by a line running from north to south. The lion is then either in the eastern or in the western part.
Lets assume it is in the eastern part.
Divide this part by a line running from east to west. The lion is either in the northern or in the southern part. Lets assume it is in the northern part. We can continue this process arbitrarily and thereby constructing with each step an increasingly narrow fence around the selected area. The diameter of the chosen partitions converges to zero so that the lion is caged into a fence of arbitrarily small diameter.