1. Show that lim (n^n)/(n!e^n) exists without finding the limit.

n->oo

Show that your sequence is monotone descending, which is surprisingly easy if you already know that the sequence

\(\displaystyle \displaystyle{\left(1+\frac{1}{n}\right)^n\) converges monotonically __ascending__ to \(\displaystyle e\)

2. Show that lim (1/n){(2*4*...*(2n))/(1*3*...*(2n-1))}^2 exists without finding the limit. n->oo

I guess I have to prove this by least upper bounds, however hard to start!

You can try here the above, but it'll be much more involved. First, be sure you can show that

\(\displaystyle \displaystyle{2\cdot 4\cdot\ldots\cdot (2n)=2^n\,n!\,,\,\,1\cdot 3\cdot\ldots\cdot (2n-1)=\frac{(2n)!}{2\cdot 4\cdot\ldots\cdot (2n)}}\) , so

that now you can write your sequence as \(\displaystyle \displaystye{a_n:=\frac{1}{n}\left(\frac{(2^n\,n!)^2}{(2n)!}\right)^2\) , and

now show that \(\displaystyle a_{n+1}\leq a_n\) ...__very carefully and slowly!__

Tonio