Let $x, y \in \Bbb Z$. If $x + y \geq 135$, then $x > 67$ or $y > 67$.

Question

Let $x, y \in \Bbb Z$. If $x + y \geq 135$, then $x > 67$ or $y > 67$.

How do I prove this statement? I'm new to proofs, and I find this to be too obvious to prove.

Answer 1

We need Proof by contradiction

If both $x,y\le67, x+y\le67+67<135$

Answer 2

Proof: We are goning to prove that $x>67$ or $y>67$. Now we assume that both $x$ and $y$ satisfy that $x \le 67$ and $y \le 67$. So $x+y \le 67+67<135$. This is a contrddiction. So we have that $x>67$ or $y>67$.