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Homework Statement
Let K be a nonempty compact set in R2Prove that the following set is compact:
[tex]S=\lbrace{p\in R^{2}:\parallel p-q\parallel\leq 1 for some q\in K}\rbrace [/tex]
Homework Equations
I will apply Heine-Borel- i.e. a set is compact iff it is bounded and closed
The Attempt at a Solution
This is somewhat of a silly question, I know that all I have to do is split the set up into too compact sets A and B (i.e. A+B=S), I then already know how to prove that the sum of two nonempty compact sets is compact. The problem is, I am having trouble splitting the set up ( I am still fairly new to set notation, which doesn't help).
So far I have looked at
[tex]A=\lbrace {p:\parallel p \parallel \leq R, p\in R^{2}\rbrace[/tex]
This set is both closed and bounded and therefore compact. and...
[tex]A=\lbrace {q:\parallel q \parallel \leq 1-R, q\in K\rbrace[/tex]
This set is also bounded and closed.
If I add the two sets I get the right condition through the triangle inequality, but I clearly end up with
[tex]A+B=\lbrace {p+q:\parallel p-q \parallel \leq 1 \rbrace[/tex]
which isn't what I want,
I think more than anything I am just struggling with the set operation, any help would be much appreciated- I have done the hard(er) part of the proof, my understanding is that this part should be simpler, but I am struggling.
Thanks
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