general topology – Can we prove that $mathbb{Q} subseteq mathbb{R}$ is $T_4$

Suppose that $mathbb{Q} subseteq mathbb{R}$ equipped with the subspace (Euclidean) Topology is a $T_4$ space.

So my thoughts go as follows: We know that $mathbb{Q}$ is a subspace, so if we can show that $mathbb{R}$ is metrizable (I think with the Euclidean topology) then we get that $mathbb{Q}$ is $T_4$.

If my thought process is correct, do I still need to show that $mathbb{R}$ is metrizable w.r.t the Euclidean topology.