elementary set theory - Cardinality of all injective functions from $mathbb{N}$ to $mathbb{R}$.

What is the cardianlity of: $$ A = \left\{ f:\mathbb{N}\to\mathbb{R} : \text{f is injective} \right\} $$

Trying to prove it using Cantor–Bernstein–Schroeder theorem, I have the obvious side:
$$A \subseteq f:\mathbb{N}\to\mathbb{R}$$

Hence,
$$\left|A\right| \le \aleph$$

I need to find an injection from a set with cardinality of $\aleph$ to $A$, but couldn't think of a proper one. It's tricky.

Any idea?

Thanks.

Answer

HINT: Prove that $\{f\in A\mid\operatorname{range}(f)\subseteq\Bbb N\}$ has size $\aleph$.