Expressing "For every positive integer $a$, there exists an integer $b$ with $|b| < a$ such that $|bx| < a$ for every real number $x$" symbolically

Question

I would like to express

For every positive integer $a$, there exists an integer $b$ with $|b| < a$ such that $|bx| < a$ for every real number $x$

symbolically. My attempt:

For every positive integer $a$

This tells us that $\forall a \in \mathbb{Z}^+$.

there exists an integer $b$

This tells us that $\exists b \in \mathbb{Z}$

with $|b| < a$

This tells us that $P(a,b) : |b| < a$.

such that $|bx| < a$ for every real number $x$

This tells us that $Q(a,b,x) : |bx| < a$ and $\forall x \in \mathbb{R}, Q(a,b,x)$.

Now putting it all together, I think we would have:

$\forall a \in \mathbb{Z}^+, \exists b \in \mathbb{Z}, P(a,b) \implies \forall a \in \mathbb{Z}^+, \exists b \in \mathbb{Z},\forall x \in \mathbb{R}, Q(a,b,x).$

Alternatively, it could also be

$\forall a \in \mathbb{Z}^+, \exists b \in \mathbb{Z},\forall x \in \mathbb{R}, P(a,b) \implies Q(a,b,x)$

It could also be the case that neither of them are correct I suppose. Please let me know which you think is the most correct in this case, and why. Thank you!

Answer 1

Welcome on Math SE!

What you proposed is quite good. Nevertheless, let's get back on it, identify then correct the mistakes. (NB : There different solutions that can be accepted for such question, I'm going to try the one that is best understandable.)

For every positive integer $a$

You said $\forall a\in\mathbb{Z}_+$. But $\mathbb{Z_+}$ has a "name", it is $\mathbb{N}$. So, let's write: $$\forall a\in\mathbb{N}$$

There exists an integer $b$

I will use this one with the following one and combine them:

There exists an integer $b$ with $\vert b\vert < a$.

Then, if $b\in\mathbb{Z}$ and $\vert b\vert < a$, we necessarily have $b\in]\!]-a,a[\![$. So, we can write it as: $$\exists b\in]\!]-a,a[\![$$

such as for every real number $x$, $\vert bx\vert<a$

You got it : $$\forall x\in\mathbb{R}\, (\vert bx\vert<a)$$

Now, let's put the pieces back together: $$\forall a\in\mathbb{N}\exists b\in]\!]-a,a[\![ (\forall x\in\mathbb{R}, \vert bx\vert<a)$$

If you have any questions, do not hesitate.