I want to show directly that $\mathbb{R}$ is $T_1$, meaning that for any two points $x,y \in X$ there exists two open sets $U$ and $V$ such that $x \in U$ and $y \notin U$, and $y \in V$ and $x \notin V$.
I am aware that I just need to show that one-point sets are closed in $\mathbb{R}$ and my proof will be complete.
Will someone help me write a rigorous proof for this?
Let $x,y\in \mathbb{R}, x\neq y$. Take $r<\frac{|x-y|}{2}$. Then you can take $U=B(x,r)$ and $V = B(y,r)$. This proves even more: $\mathbb{R}$ is $T_2$.