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Math Help - Real analysis homework help!!!!!!

  1. #1
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    Real analysis homework help!!!!!!

    I need help with my homework!!! I don't understand these at all and any help would be appreciated.

    1.) Prove that any finite set has a max and min.


    2.) a.) Prove:If x and y are real numbers with x<y, then there are infinitely many rational numbers in the interval [x,y]

    b.) Repeat part (a) for rational numbers.
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  2. #2
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    Quote Originally Posted by trojanlaxx223 View Post
    I need help with my homework!!! I don't understand these at all and any help would be appreciated.

    1.) Prove that any finite set has a max and min.


    2.) a.) Prove:If x and y are real numbers with x<y, then there are infinitely many rational numbers in the interval [x,y]

    b.) Repeat part (a) for rational numbers.
    1. Suppose for contradiction that some finite set  S did not have a max and min. What would happen? If there was no max for example, we could choose keep choosing larger elements that are still in  S (by Archimedean property). But this would be an infinite set with cardinality  \aleph_0 . Contradiction.


    Also for 2(b) you mean irrational numbers?
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  3. #3
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    yes irrational numbers for part b. sorry.
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  4. #4
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    2(a) is equivalent to showing that if there are a finite number of rational numbers in  [x,y] then  x > y e.g. the empty set. Or you can do a proof by contradiction. Suppose that there are a finite number of rational numbers in  [x,y] . Call this  S = \{q \in \mathbb{Q}: x \leq q \leq y \} . Since this is finite we can choose two elements without finding something in between. Call these  a and  b . But  \frac{a+b}{2} \in \mathbb{Q} . Also  \frac{a+b}{2} \in [x,y] which is in between  a and  b . Contradiction.
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