Show that every polygon is limited.

Question

I've already set polygon , polygonal , limited sets . But I have no idea where to start, tried by reductio ad absurdum but did not. Any idea?

Answer 1

Let the vertices be $v_1$ to $v_n$, and let $M$ be the maximum of the norms of the $v_i$. Let $u$ and $v$ be consecutive vertices. Then the points on the line segment joining $u$ and $v$ have the shape $$(1-t)u+tv,$$ where $0\le t\le 1$. By the Triangle Inequality, $$|(1-t)u+tv|\le (1-t)|u|+t|v|\le M.$$

Answer 2

By limited do you mean bounded, that is, enclosed in some circle (maybe centered on the origin)? You could compute the distance from the origin to each point, take the maximum, and claim the polygon is within a circle of that radius. Can you show that it is?