The truth value of $(P):(\exists m \in \mathbb{Z}) (\forall y\in \mathbb{Q}) : my\in\mathbb{N}$

Determine the truth value of the following statement: $$(P):(\exists m \in \mathbb{Z}) (\forall y\in \mathbb{Q}) : my\in\mathbb{N}$$

my attempt:

P is false, because suppose that exists $m\in\mathbb{Z}^{*}_{-}$ such that: $$(\forall y\in \mathbb{Q}) : my\in\mathbb{N} $$ let's take for example: $y=\frac{m}{m}\in\mathbb{Q}$ we got : $$m.y=m.\frac{m}{m}=m\notin \mathbb{N} $$ since $m\in\mathbb{Z}^{*}_{-}$. Therefore, the statement (P) is false.

Since you include $0\in\Bbb N$, $(P)$ is true because we have $\Bbb Z\ni 0=0y\in\Bbb N$ for all $y\in\Bbb Q$.

Your proof is wrong even if $0\notin\Bbb N$, since you only exclude negative $m$.

To prove $(P)$ is false when $0\notin\Bbb N$, let $y=0$.