first of all, I want to say that I am pretty new to math. So, if there are any mistake, please told me.
I attempt to proof that $e^{e}$ is irrational.
$e^{e} = e^{1}*e^{1}*e^{1/2!}*e^{1/3!}*...$
Suppose an integer "a" We can rewritten equation as
$e^{e} = e^{1}*e^{1}*e^{1/2!}*e^{1/3!}*...*e^{1/a!}*e^{\sum_{n=1}^{\infty} \frac{1}{(a+n)!}}$
$(e^{e})^{a!} = (e^{1}*e^{1}*e^{1/2!}*e^{1/3!}*...*e^{1/a!})^{a!}*(e^{\sum_{n=1}^{\infty} \frac{a!}{(a+n)!}})$
Induce the Limit, We got
$\lim_{a\rightarrow \infty} {(e^{e})^{a!}} = \lim_{a\rightarrow \infty}(e^{1}*e^{1}*e^{1/2!}*e^{1/3!}*...*e^{1/a!})^{a!}*(e^{\sum_{n=1}^{\infty} \frac{a!}{(a+n)!}})$
We can derive that
$(e^{e})^{integer} = e^{integer}*(e^0)$
There was a proof already that $e^{integer}$ and this also imply that $e^{e}$ is irrational because, irrational only come irrational * irrational.
But still, I'm not sure that could we use limit in this kind of proof. Thank you in advance.