|Apr19-05, 11:43 PM||#1|
Prove D U D' is bounded
The homework question is this:
Prove If D is a bounded subset of R then D bar = D U Dí is also bounded where Dí is the set of accumulation points of D.
What is a general outline of a proof?
|Apr20-05, 12:34 AM||#2|
It suffices to show that D' is bounded, as the union of 2 bounded sets is bounded.
If D is bounded, then it is contained in some finite interval, ( - N, N ). If D' is not bounded, then we can find an element, x, of D' outside of ( -N, N ), and taking a suitable neighborhood around x ( of radius less than |x| - N ), we see that it is disjoint from D ( as it is disjoint from ( -N, N ) ). Therefore, x is not an accumulation point of D. Contradiction
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