I am being asked to prove that the set { $$x \in \mathbb{R}: x^2 \le x$$ } is equal to the interval [0,1].

Now I can see that any x on that interval this is true, but when they ask me to prove this, should I be using things like induction or contradiction? Or, is this provable just using the properties of an ordered field?

Question

I am being asked to prove that the set { $$x \in \mathbb{R}: x^2 \le x$$ } is equal to the interval [0,1].

Now I can see that any x on that interval this is true, but when they ask me to prove this, should I be using things like induction or contradiction? Or, is this provable just using the properties of an ordered field?

anonymous · Answer

$$x^2\leq x\iff x^2-x\leq 0\iff x(x-1)\leq 0$$

anonymous · Answer

i am not sure how fancy you need to be, but it is pretty obvious that if $x<0$ we know $x-1<0$ and so $x(x-1)>0$

anonymous · Answer

Ah right!

anonymous · Answer

And the same holds in the other direction for x > 1

anonymous · Answer

whereas if $x>0, x-1<0$ i.e. if $0<x<1$ we have $x(x-1)<0$

anonymous · Answer

yes it does

anonymous · Answer

Thanks a ton for this, for me it seems the hardest part of a proof is getting started. I appreciate the direction.

anonymous · Answer

yw