# Thread: Proof of a maximum and minimum

1. ## Proof of a maximum and minimum

A function f is said to be increasing on a set S if the inequality f(t) is less than or equal to f(x) holds whenever t and x belong to S and t is less than or equal to x. Prove that every increasing function on the interval [0,1] must have both a max and a min.

2. What can you say about f(1)? And what can you say about f(0)?

3. Originally Posted by Slazenger3
A function f is said to be increasing on a set S if the inequality f(t) is less than or equal to f(x) holds whenever t and x belong to S and t is less than or equal to x. Prove that every increasing function on the interval [0,1] must have both a max and a min.
Do you want to say that the max/min occurs on the interval?

4. I believe so...

5. Then look again at JoachimAgrell's response!

6. Originally Posted by Slazenger3
I believe so...
To give a further hint:

Take any number in $\displaystyle [0,1]$, say $\displaystyle c$. Now compare $\displaystyle f(c)$ to $\displaystyle f(0)$ and $\displaystyle f(1)$.