Prove that every uncountable closed subset of $\mathbb{R}$ has cardinality $2^{\aleph_0}$

Question

The continuum hypothesis states that there is no set $X$ such that $\aleph_0<|X|<2^{\aleph_0}$. In this problem we have to prove that the continuum hypothesis holds for closed subsets of $\mathbb{R}$. In order to prove that statement, we can show that there is an injective function $f$ from $\mathbb{R}$ into $A$, and then apply the Cantor-Bernstein Theorem. However, I have not been able to prove the existence of this function.