How to prove $\forall x, x^2 \gt x$?

Question

I can't figure out how to prove $\forall x, x^2 \gt x$? I tried substituting $x$ with $2k+1$ and I got $4k^2>-2k$. Besides, I also have problem proving $\forall x,x>1→x^2>x$. Any help will be appreciated.

Answer 1

HINT

I think you are making things more difficult for yourself with that substitution.

$x^2\gt x\Rightarrow x^2-x\gt 0 \Rightarrow x(x-1)\gt0$

Is this always true?

Answer 2

1) Let $1 \lt x.$

Multiply the above inequality by $x (\gt 0):$

$\rightarrow$ $x \lt x^2.$

2) Let $0 \lt x \lt 1.$

$\rightarrow$ $x^2 \lt x.$

3)Let $x \lt 0.$

Is $x^2 \gt x$ true?