Let's say we have two injective functions $u : \mathbb{N} \to A$ and $v : A \to \mathbb{N}$.
How can I demonstrate that $A$ is uncountable ?
If there is an injective function $u : \mathbb{N} \to A$ and $v : A \to \mathbb{N}$, consequently there is a bijection between $\mathbb{N}$ and $A$. But what can I do from that?
Thanks!
Either you've misread something or there's a typo in the exercise. If there is an injection from $A$ to $\mathbb{N}$ then $A$ is countable; in fact, this is the definition of "countable"! (Actually, your text might define countable as "in bijection with $\mathbb{N}$;" the difference is whether we decide that finite sets are countable. Note that in this problem, the injection from $\mathbb{N}$ to $A$ ensures that $A$ is infinite, so whichever definition you're using works.)