A quantified implication

#forall implies#existential#mixed

Statement

Prove: $\forall x\in\mathbb{R},\ (x>1\Rightarrow x^2>x)$ .

Hint

Use direct proof: assume $x>1$ and multiply by $x>0$ .

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Proof. Let $x\in\mathbb{R}$ with $x>1$ . Since $x>0$ , multiplying $x>1$ by $x$ gives $x^2>x$ . ✓

Conclusion. For every real $x>1$ , $x^2>x$ .