Prove that any finite set has a maximum and a minimum.
I know that I have to prove this using mathematical induction...
I mean a finite set of real numbers.
I can use the field axioms, order axioms, absolute value, the completeness axiom, archimedean property, and density of the rationals to prove this.
I know I need to know that such a finite set of real numbers must be bounded above. I know I need to use the completeness axiom so that the set has a least upper bound. I think that I need to show that the LUB belongs to the set??? I just cant pull it all together...
HOW do you know there is a lub? Wouldn't you have to prove that a finite set has an upper bound first? I don't see that you need to use completeness on a finite set. Use induction on the size of the set:
Given two numbers, a, b, it is simple to determine which is larger and which is smaller. That is min(a,b) and max(a,b) always exists.
If S is contains a single number then that number IS both the max and min.
Assume that every set with k members has both a max and min and let S be a set with k+1 points. Let a be any member of S. Then S\{a} contains n-1 points and so has max and . Then the max of S is max{ , a} and the min of S is min , b}.