Show a nonempty finite subset of R has a max and min

Show that a nonempty finite subset of the Real Numbers has both a maximum and minimum.

I know I can use induction to solve this but I seem to be having some problems getting started:

Let P(n) be assertion: a nonempty finite subset of R has a max and min

P(1) is the assertion: a set S={1} has a max and min, both of which are 1.

Suppose P(k) is true for: a set S={k} containing a max and min.

Show P(k+1) true for S={k+1}

This is where I seem to be getting stuck. I am just not sure what to do from here (or even if my set up is correct)

Any help would be greatly appreciated!

Thank you :D

Re: Show a nonempty finite subset of R has a max and min

Re: Show a nonempty finite subset of R has a max and min

Now there are a few cases to cover!

Here's what I have come up with now,

Let and

if then is a maximum for T

if then is a maximum for T

if then is a minimum for T

if then is a minimum for T

Re: Show a nonempty finite subset of R has a max and min

Quote:

Originally Posted by

**mybrohshi5** Now there are a few cases to cover!

Here's what I have come up with now,

Let

and

if

then

is a maximum for T

if

then

is a maximum for T

if

then

is a minimum for T

if

then

is a minimum for T

You are trying to show that has a max & min.

is the set in the inductive step and has one more element.

Re: Show a nonempty finite subset of R has a max and min

Okay I may be getting a little confused now.

so i have a set S which contains {n+1} elements. Now I let be an element of S and let T = S \ {x}.

So now T is of size "n" and we know that T contains a max and min, namely and

if then is a maximum for S

if then is a maximum for S

similar reasoning for minimum

Thus we have shown that S will always contain a max and min (whether that be x, or )

Am I missing something?

Re: Show a nonempty finite subset of R has a max and min

Quote:

Originally Posted by

**mybrohshi5** so i have a set S which contains {n+1} elements. Now I let

be an element of S and let T = S \ {x}.

So now T is of size "n" and we know that T contains a max and min, namely

and

if

then

is a maximum for S

if

then

is a maximum for S

similar reasoning for minimum

Thus we have shown that S will always contain a max and min (whether that be x,

or

)

**Am I missing something?**

No, you have it now. You may want to add that and

Re: Show a nonempty finite subset of R has a max and min

Great! I will do that.

Thanks again Plato. Your knowledge is greatly appreciated. Have a great day!