I'm trying to understand a step from a proof of the Stolz-Cesaro Theorem.
Let ${\left\{ {{b_n}} \right\}_{n \in {\Bbb N}}}$ is a positively strictly increasing unbounded sequence.
If ${\left\{ {{a_n}} \right\}_{n \in {\Bbb N}}}$ is another sequence and
$$\mathop {\lim }\limits_{n \to \infty } \frac{{{a_{n + 1}} - {a_n}}}{{{b_{n + 1}} - {b_n}}} = l $$
then
$$\mathop {\lim }\limits_{n \to \infty } \frac{{{a_n}}}{{{b_n}}} = l$$
The proof I'm trying to understand is here. I can't seem to understand the last step, i.e. why: $$(l-\epsilon)(1-\frac{b_{N(\epsilon)}}{b_{k+1}})+\frac{a_{N(\epsilon)}}{b_{k+1}} < \frac{a_{k+1}}{b_{k+1}}<(l+\epsilon)(1-\frac{b_{N(\epsilon)}}{b_{k+1}})+\frac{a_{N(\epsilon)}}{b_{k+1}} \implies\\ \implies l-\epsilon<\frac{a_{k+1}}{b_{k+1}} < l + \epsilon$$
How is that true and how can one write that more formally and detailed?
Thanks in advance!