Understanding Convex Analysis: Solving a Sequence of Sets

[Constantinos]

Homework Statement

Hello!

I'm having some trouble trying to understand basic concepts of Convex Analysis (I study it independently). In particular, I have a book (Convex Analysis and Optimization - Bertsekas) which gives a definition for the convergence of a sequence of sets:

Homework Equations

where a neighborhood of a point in some R^n is any open set that includes the point.

The Attempt at a Solution

I don't really understand this. For example, imagine a sequence of concentric circles in R^2 beginning with one which has radius X_0 = 1 and each consecutive one has radius X_i = X_(i-1)+(1/2)^i . As i goes infinity, this intuitively converges to a circle with radius 2 (if I did it correctly, otherwise I believe you understand me!)

But not according to the definitions given! If I understand them, the outer limit is the circle with radius 1 (every neighborhood of every point in there has an intersection with all other circles) and the inner limit is the perimeter of the circle with radius 2 (every neighborhood of every point there has an intersection with all but finitely many of the circles). These sets are not the same. And the only way I can think them the same is if the radius stays constant all the time.

So what am I thinking wrong here?

Thanks!

 

[micromass]

If I understand them, the outer limit is the circle with radius 1 (every neighborhood of every point in there has an intersection with all other circles)

I don't understand this. Surely I can take a neighbourhood of the circle with radius 1, such that it only intersects the circle of radius 1. For example, if I take the point (1,0) on the circle, then I can take the neighbourhood B((1,0),0.1). Then this will only intersect the circle with radius 1... Or do I misunderstand something??

 

[Constantinos]

Sorry, what I meant to say is that each set of the above sequence contains all the points within each circle i.e for X_0 all points such than x^2+y^2 <= 1 and for X_1: x^2 + y^2 <= (1+ 1/2)^2 and so on... Thus, a point inside X_0, is a point of every other set and of course all its neighborhoods intersect all sets of the sequence.

Now I think I've got an idea! In the outer limit, when he says infinitely many, he doesn't mean all (they could or they could not be all). And when he says "all but finitely many" he means that the neighborhood intersects infinitely many sets (for example those close to having radius 2) but some(or none) are left out. Thus I think both the outer and the inner limit is the set which contains all the points of X_0 (I am not sure about the perimeter) and the points of the perimeter of X_infinite i.e the circle with radius 2.

What do you think?

 

[micromass]

Indeed, both the inner and outer limit will be the ball of radius 2. This contains of course the ball of radius 1.

 

[Constantinos]

That's not exactly what I meant above, but of course you are right! I didn't see how any neighbor of a point inside X_inf could have infinite intersections with the others since I took infinite to mean "all" but that was the mistake. Thanks for the reply, nice forum too by the way!