Hello Nostalgia Originally Posted by

**Nostalgia** Hello,

I need to determine whether the following statement is true of false:

$\displaystyle \forall x(x > 1\to x^2 > x)$ Domain: All reals

I think the statement is true since I cannot find a value which would make the first part true while the second part false. However, I don't know how to prove the statement true without plugging in various numbers, but doing so would not prove that the statement is true in general, so I was wondering how one would start this question.

Thanks

The proof is quite simple, provided we are given that we may multiply both sides of an inequality by a positive number; i.e. provided we know that:$\displaystyle a > b$ and $\displaystyle c > 0 \Rightarrow ac > bc$

For we simply multiply both sides by $\displaystyle x$, noting that $\displaystyle x>1 \Rightarrow x >0$. So:$\displaystyle x>1 \Rightarrow xx > 1x \Rightarrow x^2>x$

Grandad