help proving statement with quantifiers and inequalities

Question

I need help with the following: Prove that for any $y$ in $\mathbb{R}$, there exists an $x$ in $\mathbb{R}$ such that $x-7>3y$. I tried to approach it by contradiction, leaving “there exists a $y$ such that for any $x$ such that $x-7\leq 3y$, though that didn’t seem very helpful either.

Answer 1

Note that $$x-7>3y \iff x>3y+7$$

Let $y$ be an arbitrary real number, the choose $x= 3y+8$ and we have $ x=3y+8 >3y+7$

Thus for every real number $y$ there exists an $x$ such as
$x=3y+8$ which satisfies $$ x-7>3y$$

Answer 2

Let $y\in \mathbb R$ then $3y+7 \in \mathbb R$

Hence now by Archimedean property, there exist $x \in \mathbb N (\subset \mathbb R$) such that

$x > 3y+7$

Which gives existence of required $x$.