Contrapositive of: If $a$ is a real number such that $|a| < r$ for every positive real number $r$, then $a=0.$

Question

I wish to state the contrapositive of:

If $a$ is a real number such that $|a| < r$ for every positive real number $r$, then $a=0.$

First, I want to state the original statement symbolically.

$\exists a \in \mathbb{R} (\forall r \in \mathbb{R}^+ (|a| < r \implies a = 0))$

Afterwards, we then can take the contrapositive:

$\forall a \in \mathbb{R}(\exists r \in \mathbb{R}^+ (a \neq 0 \implies |a| \ge r))$

The thing I am most concerned about are the quantifiers of $a$. Please let me know what you think of my attempt.

Answer 1

You have arrived at a common misconception.

The "If $a$ is a real number..." part (and variations thereof), are almost always equivalent to the phrasing (and variations thereof).

"Let $a$ be a real number..."

Then there is in fact no quantifier to change for that part and your statement becomes:

Let $a$ be a real number such that $|a|<r$ for every positive real number $r$, then $a=0$.

Then the contrapositive is:

If $a \neq 0$, then there exists a positive real number $r$ such that $|a| \geq r$.