TITLE: Proof that one large number is larger than another large number
QUESTION [16 upvotes]: Let $a = (10^n - 1)^{(10^n)}$ and $b=(10^n)^{(10^n - 1)}$
Which of these numbers is greater as n gets large?
I believe it is $a$ after looking at some smaller special cases, but I'm not sure how to prove it.

REPLY [10 votes]: Let $x=10^n$, $y=10^n-1$.
The function $\frac {x} {\ln x}$ is increasing from $e$.
Both $x$ and $y$ are greater than $e$, so $\frac {x} {\ln x} > \frac {y} {\ln y} \Rightarrow x\ln y > y\ln x \Rightarrow 
y^x > x^y$