Prove that if $\lim _{n \to \infty}(a_{n+1}-a_n)=L$ then $\lim_{n \to \infty}\frac{a_n}{n}=L$.
My idea is: $\forall \varepsilon >0$ $\exists N_{\varepsilon}$ such that $\forall n > N_{\varepsilon}$ $$L-\varepsilon < a_{n+1}-a_n<L+\varepsilon$$ so $$|a_n - a_m|=\left|a_{n}-a_{n-1}+a_{n-1}-a_{n-2}+a_{n-2}- \cdots - a_{m} \right|$$ $$(n-m)(L-\varepsilon)<a_n-a_m < (n-m)(L- \varepsilon)$$ but I don't know how conclude this. Can someone help me?