- #1

- 836

- 13

## Homework Statement

This is a problem I am just trying to do myself to work out some other problem.

I am trying to prove: f:[a, b] → [a, b]

Given: f is continuous on [a, b], for all x in [a, b] then df/dx < 1 , f(a) ≥ a , f(b) ≤ b.

**2. The attempt at a solution**

First I proved that f(b) - f(a) ≤ b - a. It is simple, but I can give the proof if you wish.

Now I need to prove that for all x in [a, b] , f(a) ≤ f(x) ≤ f(b), but I have no clue how to do this. I thought of using the Mean Value Theorem somehow, but I don't quite know how. I also thought of showing that f(x) ≤ f(x + dx) for x in [a, b], but I don't know how to do that either. Any help ?