Proving [Subsets, interior, open ball]

[Design]

Homework Statement

Prove that if A is a subset of B then int(A) is a subset of int(B).
int(A) = interior of A
int(B) = interior of B

The Attempt at a Solution

Take some y E int(a) , this implies that B(r,y) is a subset of A.
Given that A is a subset of B, we know that B(r,y) is a subset of B.
Now take some z E int(b), this implies that B(r,z) is a subset of B.

I got this much but I don't understand how even if I show that B(r,y) is in B(r,z), this shows that it is an int(A) is a subset of int(B) or am I totally on the wrong track?


thank you

 

[fzero]

What does it mean precisely for z to be in int(B)? Is this condition satisfied by y in int(A)?

 

[Design]

What does it mean precisely for z to be in int(B)? Is this condition satisfied by y in int(A)?

It means that there exist a B(r,z) in B. Don't understand what you mean here

 

[fzero]

It means that there exist a B(r,z) in B. Don't understand what you mean here

OK, in your original post you showed that for y in int(A), there is a B(r,y) in B. Is there a y that is not in int(B)?

 

[Design]

OK, in your original post you showed that for y in int(A), there is a B(r,y) in B. Is there a y that is not in int(B)?

No there is no y that is not that is not in int(B) since A is a subset of B.

 

[fzero]

No there is no y that is not that is not in int(B) since A is a subset of B.

Well I'd be a bit more careful here. $$A \subset B$$ is obviously important, but the crucial condition for an object y to be in int(B) is that there is a B(r,y) in B. You have all of the results that you need to finish the proof, you just need to put them in the right order.

 

[Design]

Well I'd be a bit more careful here. $$A \subset B$$ is obviously important, but the crucial condition for an object y to be in int(B) is that there is a B(r,y) in B. You have all of the results that you need to finish the proof, you just need to put them in the right order.

Thanks I think I understand what you mean.

How should I piece together the last part about how int(A) is a subset of int(B).
Should i say since the ball came from int(A) it must follow that int(A) is a subset of int(B)?

 

[fzero]

Thanks I think I understand what you mean.

How should I piece together the last part about how int(A) is a subset of int(B).
Should i say since the ball came from int(A) it must follow that int(A) is a subset of int(B)?

You've shown that every element of int(A) is also in int(B). Use the definition of a subset.