Completely regular space is hereditary

Let $X$ be completely regular, and let $Y\subseteq X$ be a subspace with the relative topology. Suppose that $F\subseteq Y$ is closed in $Y$ and $x\in Y\setminus F$. Then $x\notin\operatorname{cl}_XF$, so by complete regularity of $X$ there is a continuous $f:X\to[0,1]$ such that $f(x)=0$ and $f[\operatorname{cl}_XF]=\{1\}$. Now let $g=f\upharpoonright Y$, the restriction of $f$ to $Y$; $g:Y\to[0,1]$ is continuous, $g(x)=0$, and $g[F]=\{1\}$. Thus, $Y$ is completely regular.