Uniformly distributed subsequence

Question

Let $(x_{n})_{n\geq 1}$ be a sequence of real numbers uniformly distributed mod 1. If $E$ is a subset of $\mathbb{N}$ where the limit $$d(E)=\lim_{N\to\infty}\frac{1}{N}|\{1, 2, \dots, N\}\cap E|$$ exists and positive, then is it true that the subsequence $(x_{n})_{n\in E}$ is also uniformly distributed? Clearly it is true for $x_{n}=n\alpha$ and $E=m\mathbb{N}$ where $\alpha$ is irrational number and $m\in \mathbb{N}$. I tried to show this by using Weyl's criterion, but I can't prove it.