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

1. ## 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.

2. 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 $S$ 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?

3. yes irrational numbers for part b. sorry.

4. 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.