QUESTION
Let $\left\{x_{n}\right\}_{n}$ and $\left\{y_{n}\right\}_{n}$ be positive sequences. We say that $\left\{x_{n}\right\}_{n}$ is asymptotically similar to $\left\{y_{n}\right\}_{n}$ if $\lim _{n \rightarrow \infty} \frac{x_{n}}{y_{n}}$ exists and is non-zero. Prove each of the following true statements.
IF $\left\{x_{n}\right\}_{n}$ is asymptotically similar to $\left\{y_{n}\right\}_{n}$ THEN there exists $C>0$ such that for all $n$ large enough \[ C y_{n}<x_{n}<2 C y_{n} \]
Define the sequence $\left\{a_{n}\right\}_{n=0}^{\infty}$ by \[ \begin{aligned} a_{0} & =0, \\ \forall n \in \mathbb{N}^{+}, \quad a_{n} & =\ln (n !)-\left(n+\frac{1}{2}\right) \ln n+n . \end{aligned} \] (a) Show that IF $\left\{a_{n}\right\}_{n}$ converges THEN $n$ ! is asymptotically similar to $n^{n+\frac{1}{2}} e^{-n}$.
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