I started studying multivariable caclulus, and am having problems with this exercise:
Given an infinite bounded set A in $R^n$,$2\leq n$, show there are infinite boundary points
Attempt: Going from the assumption that A has finite boundary points, we can assume it has an interior point. I believe it's possible to construct a ball around that point with large enough radius (assuming A is bounded by M, 3M should suffice), and prove that there's a boundary point on every line between a boundary point of that ball and the interior point we selected. This ended up being very complex and I didn't manage to work out all the details nicely, so I figure there must be something easier!