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
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?