Results 1 to 4 of 4

Math Help - some more supremums and infimums (real analysis)

  1. #1
    Newbie
    Joined
    Jan 2008
    Posts
    14

    some more supremums and infimums (real analysis)

    For any two non-empty subsets of R let us write P<=Q if , for each x in P, there is a y in Q satisfying x<=y

    (a) Show that if P<=Q then supP<= supQ (sup is short for supremum)

    (b) Give an Example to show that if P<=Q then it does not follow that infP<=infQ (inf is short for infimum)

    (c) Give an example to show that if P<=Q and if Q<=P then it does not follow that P=Q
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by alexmin View Post
    For any two non-empty subsets of R let us write P<=Q if , for each x in P, there is a y in Q satisfying x<=y

    (a) Show that if P<=Q then supP<= supQ (sup is short for supremum)
    Proof:

    Let x be an arbitrary element of P and y be the element of Q such that x \le y. Now we know that y \le \sup Q for all y \in Q, by the definition of supremum. So we have x \le y \le \sup Q. In particular, we have x \le \sup Q for all x \in P. But this means that \sup Q is an upper bound for the set P. Now, by the defintion of the supremum, x \le \sup P for all x \in P AND \sup P is less than or equal to any other upper bound for P. Thus we have \sup P \le \sup Q, as desired.

    QED
    Follow Math Help Forum on Facebook and Google+

  3. #3
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by alexmin View Post
    (c) Give an example to show that if P<=Q and if Q<=P then it does not follow that P=Q
    let P be the interval [0,1) and Q be the interval (0,1)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by alexmin View Post
    (b) Give an Example to show that if P<=Q then it does not follow that infP<=infQ (inf is short for infimum)
    Let P be the interval (1,2) and let Q be the interval (0,2)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. metric spaces and infimums
    Posted in the Differential Geometry Forum
    Replies: 10
    Last Post: November 13th 2010, 06:16 PM
  2. analysis : addition of supremums
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 24th 2009, 10:01 PM
  3. infimums and supremums
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 20th 2009, 04:25 AM
  4. infimums and supremums of Functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 19th 2009, 06:06 AM
  5. real analysis...pls help me with this!
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 23rd 2008, 01:05 AM

Search Tags


/mathhelpforum @mathhelpforum