Limit for unbounded sequence Consider $\lim a_n  = +\infty$. Let $N^* \in \mathbb{N}$. Is it correct to write $\lim a_n - a_{N^*} = \lim a_n$. 
$(a_n \in \mathbb{R})_{n \in \mathbb{N}}$ by the way 
I am in the middle of proving Stolz–Cesàro theorem 
Starting from $\lim \frac{a_{n+1} - a_n}{b_{n+1} - b_{n}} = L$, I have successfully shown that $\lim \frac{a_n - a_{N^*}}{b_n - b_{N^*}} = L$ (111)
where $\lim b_n = +\infty$ and $b_n \neq 0$ for all n, $b_n$ is strictly increasing and $\lim b_n - b_{N^*} \neq 0$. 
(111) $\implies \frac{\lim a_n - a_{N^*}}{\lim b_n - b_{N^*}} = L$
Goal is to show $\lim \frac{a_n}{b_n} = L$. 
 A: As I said in the comments, you should be careful when writing expressions like $\frac{+ \infty}{+ \infty}$. E.g., your equation
$$
\frac{\lim a_n}{\lim b_n} = L
$$
does not make sense.
However, you can adapt the expressions to get finite limits. Notice that
$$
\frac{a_n - a_N}{b_n - b_N}
= \frac{\frac{a_n}{b_n} - \frac{a_N}{b_n}}{1 - \frac{b_N}{b_n}}.
$$
Thus, you can write
\begin{align*}
\lim_n \left(\frac{a_n}{b_n} - \frac{a_N}{b_n}\right)
&= \lim_n \left[
  \frac{a_n - a_N}{b_n - b_N} \cdot \left(1 - \frac{b_N}{b_n}\right)
  \right] \\
&= \lim_n \frac{a_n - a_N}{b_n - b_N} \cdot \lim \left(1 - \frac{b_N}{b_n}\right) \\
&= L \cdot 1 \\
&= L.
\end{align*}
Here we could separate the product into two limits because in this case both limits are finite. Moreover, notice that
$$
\lim_n \frac{a_n}{b_n}
= \lim_n \left(\frac{a_n}{b_n} - \frac{a_N}{b_n} + \frac{a_N}{b_n} \right)
= \lim_n \left(\frac{a_n}{b_n} - \frac{a_N}{b_n} \right) + \lim_n \frac{a_N}{b_n}
= L + 0 = L.
$$
Once again, a crucial point here is that when we separate the limit of the sum into the sum of the limits, both limits are known to be finite.
