Sequences and Asymptotic Similarity We define a sequence by $a_0 = 0$ and $a_n = \ln(n!) - \left(n+\frac{1}{2}\right)\ln(n) + n$ for positive integer $n$. 

If we know $a_n$ converges, i.e.,

$$\ln(n!) - \left(n+\frac{1}{2}\right)\ln(n) +n \to l, \text{ for some $l$ when }n \to +\infty,$$

why do we know that $\ln(n!)$ is asymptotically equaivalent to $\left(n+\frac{1}{2}\right)\ln(n)-n$, or in other words, why is the following true: $$\lim_{n\to +\infty}\frac{\ln(n!)}{\left(n+\frac{1}{2}\right)\ln(n)-n}= L,\, \text{ for some finite } L?$$
It follows that
$$
\mathop {\lim }\limits_{n \to  + \infty } \frac{{\log n!}}{{\left( {n + \frac{1}{2}} \right)\log n - n}} = 1 + \mathop {\lim }\limits_{n \to  + \infty } \frac{{a_n }}{{\left( {n + \frac{1}{2}} \right)\log n - n}} \\ = 1 + \frac{l}{{\mathop {\lim }\limits_{n \to  + \infty } \left( {\left( {n + \frac{1}{2}} \right)\log n - n} \right)}} = 1.
$$
It follows that
$$
\mathop {\lim }\limits_{n \to  + \infty } \frac{{\log n!}}{{\left( {n + \frac{1}{2}} \right)\log n - n}} = 1 + \mathop {\lim }\limits_{n \to  + \infty } \frac{{a_n }}{{\left( {n + \frac{1}{2}} \right)\log n - n}} \\ = 1 + \frac{l}{{\mathop {\lim }\limits_{n \to  + \infty } \left( {\left( {n + \frac{1}{2}} \right)\log n - n} \right)}} = 1.
$$