This is part of the proof for the Donsker Theorem form Mörters Brownian Motion. There, the authors state, that for any sequence $(a_n)_{n\geq 1}$ with $$\lim_{n\rightarrow\infty}\frac{a_n}{n}=1$$ it is implied that
$$\lim_{n\rightarrow\infty}\sup_{0\leq k\leq n}\left|\frac{a_k-k}{n}\right|=0.$$
Intuitively, it seems clear. If $a_n/n$ converges to $1$, than the the absolute of the biggest deviation of $a_n$ from the diagonal n should be going to zero. But i have difficulties to see, how one would proof that formally. Can somebody give me a hint?