This problem is from A ProblemText in Advanced Calculus by J. M. Erdman (Chap. 16: The Heine-Borel Theorem, p. 91).
Use the Cantor intersection theorem to show that the medians of a triangle are concurrent.
This is the problem that surprised me the most in the full text. Any suggestions?
Here's the statement of the theorem we are supposed to use:
Theorem (Cantor intersection theorem). If $(A_n)$ is a nested sequence of nonempty closed bounded subsets of $\mathbb R^n$, then $\bigcap\limits_{n=1}^\infty A_n$ is nonempty. Furthermore, if $\operatorname{diam} A_n \to 0$, then $\bigcap\limits_{n=1}^\infty A_n$ is a single point.
My thoughts.
Of course, I know some standard proofs of this result (using plane geometry or vectors). But here, we are required to use the Cantor intersection theorem.
Fix an arbitrary triangle $\triangle$. I originally started out with $A_1 := \triangle$. My goal was to assume that the medians are not concurrent, and derive a contradiction. It is clear that if the medians are not concurrent, then they form a mini-triangle inside $A_1$. Should I define that triangle to be $A_2$ and proceed recursively to define $A_3, A_4, \ldots$? This idea is the only one I could think of. However, this does not seem good enough: I cannot see any contradiction arising from the conclusion that the intersection of all the $A_n$'s is nonempty (or a singleton set).
Can you suggest a better approach?