In fact, we have for the bases of the individual triangles the value 2 sin p/n and for the height, by Pythagoras' theorem,
Since the number of triangles is obviously 2mn, the surface area of the polyhedron is
The limit of this expression is not independent of the way in
which m and n tend to infinity. For example, if we
keep n fixed and let m ®
¥ , the expression increases beyond all bounds.
However, if we make m and n tend to ¥ together, setting m=n, the expression tends to 2p.
If we put m = n², we obtain the limit 
,
etc. We see from the above expression Fn,m
for the area of the polyhedron that the lower limit (lower point of accumulation)
of the set of numbers Fn,m is 2p ; this follows at once from Fn,m
³ 2p
sin p/n and 
l
In conclusion, we mention - without proof - a theoretically interesting fact of which the example just given is a particular case. If we have any arbitrary sequence of polyhedra tending to a given surface, we have seen that the areas of the polyhedra need not tend to the area of the surface. But the limit of the areas of the polyhedra (if it exists) or, more generally, any point of accumulation of the values of these areas is always larger than, or at least equal to the area of the curved surface. If we find for every sequence of such polyhedral surfaces the lower limit of the area, these numbers form a definite set of numbers associated with the curved surface. The area of the surface can be defined as the lower limit (lower point of accumulation) of this set of numbers.
This remarkable property of the area is called semi-continuity, or more precisely lower semi-continuity.