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Hi how to show the following:

Let $C$ and $D$ be two non-empty closed convex sets with no common direction of recession. Then $C - D$ is closed.

Thanks a lot...

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up vote 2 down vote accepted

Chances are that your question would not be ignored for a week if you included your own thoughts on this problem.

Suppose $C-D$ is not closed, what would that imply? We would have a convergent sequence of points in $C-D$ the limit of which is not in $C-D$. The terms of this sequence are $c_n-d_n$, with $c_n\in C$ and $d_n\in D$. Let $a$ be its limit.

If $c_n$ had a convergent subsequence $c_{n_k}\to c$, then $d_{n_k}$ would also converge (why?), and since $C$ and $D$ are closed, we would find that $a\in C-D$. Therefore, both sequences go to infinity.

Consider the sequences of unit vectors $c_n/|c_n|$ and $d_n/|d_n|$. Observe that $$\frac{c_n}{|c_n|}- \frac{d_n}{|d_n|}\to 0 \tag{why?}$$ Since the unit sphere is compact, we have a convergent subsequence $\frac{c_{n_k}}{|c_{n_k}|}\to u$, and therefore its counterpart converges to $u$ as well. This vector $u$ provides you with a common direction of recess for both $C$ and $D$. (This is where you'll need convexity.)

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