Is it possible to define the circumference of a circle as the limit of a sequence of perimeters of inscribed polygons when the sequence of lengths of the largest sides of the polygons tend to zero.
This is problem from the book 'Elementary problems in mathematics selected topics and problem solving'.
The answer states that Impossible. For example let us take the sequence of inscribed triangles $A_nB_nC_n$ for which the arc $AB$ and $BC$ are always equal to $2π/n$ (what is the limit of the sequence of perimeters of these triangles?).
The answer ends here and I can't get it.
The answer actually means:
if we determine the limit of the perimeter of $\triangle A_nB_nC_n$, we will find the limit is $0$ (As the lengths of all three sides of the triangle tend to $0$ while $n$ tends to positive infinity), which means that the limit is not necessarily the perimeter of the circle $2\pi$.