Find the limit of $a_n = e^ne^{-e^n}$

Question

Consider the sequence $(a_n)$ where $a_n = e^ne^{-e^n}$, what is $\lim_{n \rightarrow \infty} (a_n)$?

I have a feeling it's 0 because $\displaystyle a_n = \frac{e^n}{e^{e^n}}$ and $e^n$ grows faster than $n$, but how do I show it rigorously? I'm thinking the squeeze theorem might work?

Answer 1

It's an elementary exercise to show that $e^n > 2n$ for all sufficiently large $n$, implying that

$$\frac{e^n}{e^{e^n}} < \frac{e^n}{e^{2n}} = \frac{1}{e^{n}}$$

for these values of $n$. Now squeeze.

Answer 2

$e^ne^{-e^n}=e^{n-e^n}\to0$ because $n-e^n\to-\infty$.