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## Homework Statement

I have been self studying Spivak's Calculus on Manifolds, and in chapter 1, section 2 (Subsets of Euclidean Space) there's a problem in which you have to find the interior, exterior and boundary points of the set

[tex]

U=\{x\in R^n : |x|\leq 1\}.

[/tex]

While it is evident that

[tex]

\{x\in R^n : |x|\lt 1\},

\{x\in R^n : |x|= 1\},

\{x\in R^n : |x|\gt 1\}

[/tex]

are the interior, boundary and exterior of U, in that order, I am stuck proving it. In particular, I can't quite grasp how to prove rigorously that the set [itex] \{x\in R^n : |x|= 1\} [/itex] is the boundary of U; I need to show that if [itex] x [/itex] is any point in said set, and A is any open rectangle such that [itex] x\in A [/itex], then A contains a point in U and a point not in U. If x is such that [itex] |x|=1 [/itex], then [itex] x\in U [/itex], so I know that any open rectangle [itex] A [/itex] about the point[itex] x [/itex] contains at least one point in U (namely [itex]x[/itex]), how do I know my open rectangle [itex] A [/itex] also contains points for which [itex] |x|\gt 1 [/itex]?

## Homework Equations

An open rectangle in [itex] R^n [/itex] is a set of the form [itex] (a_1,b_1)\times ... \times (a_n,b_n) [/itex].

Spivak defines interior, exterior and boundary sets using open rectangles, not open balls.

## The Attempt at a Solution

It is obvious that the boundary of the n-ball is the n-sphere, and most books wouldn't bother proving it, but I like to be rigorous in my proofs. I am getting stuck in the technical details (how do I know not all points in my open rectangle are equidistant from the origin?, how do I know at least one is "farther away?", that kinda stuff).