A sequence of Continuous Functions Converges Uniformly over $\mathbb{R}$ if it Converges Uniformly over $\mathbb{Q}$ I'm trying to show that if ${f_n}$ is a sequence of real functions that is continuous over all of $\mathbb{R}$ and that converges uniformly to $f$ over $\mathbb{Q}$, then it converges uniformly to $f$ over $\mathbb{R}$. The hint I'm given is to use the Cauchy criterion for uniform convergence.
I know that $f$ is continuous over $\mathbb{Q}$ and that $\mathbb{Q}$ is dense in $\mathbb{R}$, but using the definitions of continuity and denseness I'm still having trouble showing that $f_n$ is uniformly Cauchy.
Any help at all would be much appreciated!

let $r\in \mathbb Q$ then by definition $|f_n(r)-f_m(r)|<\frac{\epsilon}{2}\forall n,m\geq p$
Since $\mathbb Q$ is dense in $\mathbb R$ given $x\in \mathbb R\exists r\in \mathbb Q$ such that $|x-r|<\delta $ 
Since $f_n$ is continuous $|x-r|<\delta \implies |f_n(x)-f_n(r)|\epsilon \forall n\geq p$
thus for $x\in \mathbb R,|f_n(x)-f_m(x)|=|(f_n(x)-f_n(r))+(f_n(r)-f_m(r))|\leq... $
