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Real Analysis

  1. Dec 6, 2014 #1
    1. The problem statement, all variables and given/known data
    Let ##O## be a proper open subset of ##\mathbb{R}^d## (i.e.## O## is open, nonempty, and is not equal to ##\mathbb{R}^d##). For each ##n\in \mathbb{N}## let
    ##O_n=\big\{x\in O : d(x,O^c)>1/n\big\}##
    Prove that:
    (a) ##O_n## is open and ##O_n\subset O## for all ##n\in \mathbb{N}##,
    (b) ##O_1\subset O_2 \subset \dots ##, and ##\cup_n O_n=O##
    (c) If ##O_n\neq 0## then ##d(O_n, O^c)\ge \frac{1}{n}##
    (d) If ##O_n\neq 0## then ##d(O_n, O^c_{n+1})\ge \frac{1}{n(n+1)}##

    2. Relevant equations


    3. The attempt at a solution
    (a) solved
    (b) solved
    (c) I'm not sure what the instructor is looking for here since there is no ##n## and no ##x\in O_n## such that ##d(x,O^c)=\frac{1}{n}## since that would contradict the construction of ##O_n##. It seems like the problem statement should be
    If ##O_n\neq 0## then ##d(O_n, O^c)> \frac{1}{n}##.
    (d) ##d(O_n,O^c_{n+1})=d(O_n,O^c)-d(O_{n+1},O^c)\ge \frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}##
     
  2. jcsd
  3. Dec 6, 2014 #2

    Stephen Tashi

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    The following just a suggestion, not necessarily a useful "hint":

    How do your course materials define the distance between two sets? Perhaps to find [itex] d(O_n, O^c) [/itex] you might have to do something like take a limit of distances [itex] d(x_i,O^c) [/itex] with each [itex] d(x_i,O^c) \gt \frac{1}{n} [/itex] but with the sequence of distances converging to [itex] \frac{1}{n} [/itex].
     
    Last edited: Dec 6, 2014
  4. Dec 7, 2014 #3
    The distance between the two sets is ##inf d(x_i, y_i)## where ##x_i\in O_n##, ##y_i\in O^c##. But the problem I see is that by part (a) ##O_n## is open (but bounded) hence the lower bound of ##d(O_n, O^c)## is not attainable even though ##O^c## is closed. Also, the limits don't coincide since
    ##lim_{n\to\infty} d(O_n,O^c)\to 1/n## but ##lim_{n\to\infty}1/n\to 0##
     
  5. Dec 7, 2014 #4

    Stephen Tashi

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    One need not attain an infimum for it to be an infimum. Let [itex] O [/itex] be the open interval [itex] (0,1) [/itex]. and let [itex] N = 4 [/itex]. I think [itex] O_n [/itex] is the open interval [itex] ( \frac{1}{4}, \frac{3}{4}) [/itex]. What is distance between [itex] O^C [/itex] and [itex] ( \frac{1}{4}, \frac{3}{4})[/itex] ?.
     
  6. Dec 8, 2014 #5
    OK thanks, I guess saying ##d(O_n, O^c)\ge 1/4## is just another way of writing what the lower bound is. I was probably overthinking ( possibly under-thinking) things.

    Does part (d) seem correct?
     
  7. Dec 8, 2014 #6

    Stephen Tashi

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    You should explain why ##d(O_n,O^c_{n+1})=d(O_n,O^c)-d(O_{n+1},O^c)##
     
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