What is an effective approach to proving that interior points are open?

[pokemonsters]

Homework Statement

For S \subset Rⁿ, prove that S° is open.

Homework Equations

S° are all interior points of S.

The Attempt at a Solution

My class has only learned how to use balls to solve these types of problems (no metric spaces). So I need to choose an ε > 0 so that B_ε(x) \subset S°, where x is any arbitrary point in S°. To show this is true, let y \subset B_ε(x) be arbitrary. (then I don't know how to progress further...how do I show that the neighbourhood contains only points in S°?)

 

[jbunniii]

Homework Statement

For S \subset Rⁿ, prove that S° is open.

Homework Equations

S° are all interior points of S.

The Attempt at a Solution

My class has only learned how to use balls to solve these types of problems (no metric spaces). So I need to choose an ε > 0 so that B_ε(x) \subset S°, where x is any arbitrary point in S°. To show this is true, let y \subset B_ε(x) be arbitrary. (then I don't know how to progress further...how do I show that the neighbourhood contains only points in S°?)

Start by choosing an arbitrary x \in S^o. By definition this is an interior point of S, so there exists \epsilon > 0 such that B_\epsilon(x) \subset S. Now, if you can show that every point y \in B_\epsilon(x) is an interior point of S then you're done. To do this, it certainly suffices to show that you can fit a smaller ball B_\delta(y) around y which is entirely contained within B_\epsilon(x), because then you will have y \in B_\delta(y) \subset B_\epsilon(x) \subset S. Try drawing a picture to see how to define \delta, the radius of the smaller ball.

 

[pokemonsters]

Start by choosing an arbitrary x \in S^o. By definition this is an interior point of S, so there exists \epsilon > 0 such that B_\epsilon(x) \subset S. Now, if you can show that every point y \in B_\epsilon(x) is an interior point of S then you're done. To do this, it certainly suffices to show that you can fit a smaller ball B_\delta(y) around y which is entirely contained within B_\epsilon(x), because then you will have y \in B_\delta(y) \subset B_\epsilon(x) \subset S. Try drawing a picture to see how to define \delta, the radius of the smaller ball.

Thank you, I did not notice that you can put another ball inside the ball to make the proof work.

Using this, I was able to make a series of inequalities using the triangle inequality, and managed to prove that y \in B_\delta(y) \subset B_\epsilon(x) \subset S.