I am reading this paper stuck at Theorem 2.1.4
The inequation $${1\over{q(n)-p(n)}}\sum_{k=p(n)+1}^{q(n)}|x_k-l|\le M {1\over{q(n)-p(n)}}|\{k:p(n)\lt k\le q(n) , |x_k-l|\ge \epsilon\}|\\+\epsilon{{1\over{q(n)-p(n)}}|\{k:p(n)\lt k\le q(n) , |x_k-l|\lt \epsilon\}}|$$ is established. Then it says that from the limit relations we have $$\lim_{n\rightarrow \infty}{1\over {q(n)-p(n)}}\sum_{k=p(n)+1}^{q(n)}|x_k-l|=0$$
I know the first summand of RHS tends to $0$ as $n\rightarrow \infty$ by definition of $DS[p,q]$ convergence but what ensures that the second summand also tends to $0$ as $n\rightarrow \infty \ ?$
Please help somebody.