couple of real analysis questions

[trickslapper]

1. Assume A and B are open sets. Also Assume that the closure of A and B = The Real numbers.

Prove that the closure OF A and B's intersection is also = The Real numbers.


2. Let A be nonempty and bounded above. But A has no largest element.

Prove that the (supremum) sup(A) is a limit point of A.


If someone could maybe give me a push in the right direction with these two or show me how to prove them it would totally help me to study for an upcoming test. Thanks !

 

[daon]

For the second one, Let S = sup of A.

First show that B(n) = N(1/n, S) \ N(1/(n+1), S) is is nonempty and intersects with A. Show also that B(n+1), B(n) are disjoint.

Then we identify any point in A intersect B_n as a "representative" of the set, since B(n) is disjoint from B(n+1) this forms an equivalance class of points and assignments can be made uniquely.

Let b_n be the point picked in B(n).

Show the sequeince {b_n) converges to S, and conclude that since b_n are in A for every n, S is a limit point of A.

 

[trickslapper]

Hey sorry, but when you write B(n) what are you doing there? Are you creating a set B using N(1/n, S) \ N(1/(n+1), S)? If so, why (and how did you know to do this)?
And to show that B(n) and B(n+1) are disjoint, i just set them equal to each other and show that they are not?

And when you write B_n what does that stand for, i know later in your proof you say that b_n is a point in B, but what is B_n?

Thanks!

 

[daon]

I just missed the parenthesis, I meant B(n), and that b_n belongs to B(n).

The first thing that always comes to my mind in sequences is the distance from the limit each point has. When I need to come up with a sequence that shows convergence, I look at the space and how "balls" of different sizes around a limit point intersected with the space might look. This case was easy, it gets harder when sets are defined in "interesting" ways.

For fun, try to show that every point in the cantor set is a limit point of the set.