Real Analysis Proof: any finite set has maximum, minimum

[bearej50]

Prove that any finite set has a maximum and a minimum.

I know that I have to use mathematical induction to prove this...

 

[pka]

Re: Real Analysis Proof

BTW: I am sure that you meant “finite set of real numbers”.
Without knowing the exact sequence of axioms and proven theorems you have working for you, it is impossible to offer a proof of this. You need to know that such a finite set of real numbers must be bounded above. You need some sort of completeness axiom so that the set has a least upper bound. Then you show that the LUB belongs to the set.

 

[bearej50]

Re: Real Analysis Proof

I can use the field axioms, order axioms, absolute value, the completeness axiom, archimedean property, and density of the rationals

 

[PAULK]

Prove that any finite set has a maximum and a minimum.

I know that I have to use mathematical induction to prove this...

If you know that, it should be easy. A finite set (of reals, we assume) means { x[1],..x[n] }

A maximum means an element x[k] of the set having the property that ....

It is not the same as an upper bound, although it is one.

So if every set of k elements has a maximal element x[j], can you prove that a set { x[1],...x[k], x[k+1] } has one?

[If you are not sure what to do, see the thread entitled HORSE OF ANOTHER COLOR in the Odds and Ends section of this site. It's a beautiful example of inductive proof (if I say so myself).]