Can anybody show me the following Proof?
Prove that if $X_n$ for $n=1,2,\ldots,\infty$ is a real sequence that is uniformly distributed modulo $1$, and if $Y_n$, $n=1,2,\ldots,\infty$ is a real sequence such that $\lim_{n\to\infty} X_n – Y_n = \alpha$ then $Y_n$ is uniformly distributed modulo $1$.
Remark: This includes the situation where $X_n – Y_n = \varepsilon$ tends to zero as $n$ tends to infinity.
Hint: Prove for zero case first. Show that $X_n$ and $Y_n$ are close.