# Real analysis homework help!!!!!!

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• Feb 27th 2009, 05:02 PM
trojanlaxx223
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.
• Feb 27th 2009, 05:21 PM
manjohn12
Quote:

Originally Posted by trojanlaxx223
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 $\displaystyle S$ 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 $\displaystyle S$ (by Archimedean property). But this would be an infinite set with cardinality $\displaystyle \aleph_0$. Contradiction.

Also for 2(b) you mean irrational numbers?
• Feb 27th 2009, 05:32 PM
trojanlaxx223
yes irrational numbers for part b. sorry.
• Feb 27th 2009, 05:36 PM
manjohn12
2(a) is equivalent to showing that if there are a finite number of rational numbers in $\displaystyle [x,y]$ then $\displaystyle x > y$ e.g. the empty set. Or you can do a proof by contradiction. Suppose that there are a finite number of rational numbers in $\displaystyle [x,y]$. Call this $\displaystyle S = \{q \in \mathbb{Q}: x \leq q \leq y \}$. Since this is finite we can choose two elements without finding something in between. Call these $\displaystyle a$ and $\displaystyle b$. But $\displaystyle \frac{a+b}{2} \in \mathbb{Q}$. Also $\displaystyle \frac{a+b}{2} \in [x,y]$ which is in between $\displaystyle a$ and $\displaystyle b$. Contradiction.