I've been studying the real number system from Apostol Calculus Vol I and the theorem I.31 says the following:
If three real numbers a, x and y satisfy the inequalities $a \leq x \leq a + \frac{y}{n} $ for every $n \geq 1$, then $x = a$
Now, i've tried to prove the theorem by contradiction and in my head (i think i'm wrong by the way) the statement using quantifiers is
$\forall x \in \mathbb{R} \, \forall y\in \mathbb{R} \, \forall a \in \mathbb{R} \, \forall n \geq 1 \, (a \leq x \leq a + \frac{y}{n}) \implies (x = a)$
and so the negation is
$\exists x \in \mathbb{R} \, \exists y\in \mathbb{R} \, \exists a \in \mathbb{R} \, \exists n \geq 1 \, (a \leq x \leq a + \frac{y}{n}) \wedge (x \neq a)$
But then i saw the proof in the book and Apostol uses the Archimedean property to show that if $x > a$ then there exists a number $n$ such that $n(x-a) > y \,$ and $x > a + \frac{y}{n}$. And well, this doesn't match my intuition for the statement nor the negation of the statement. What would be the correct quantified statement and its negation?