Prove that $a_n
So I got this question and I was able to do up to (III) (a) however part b has been pretty difficult for me to complete and I can't seem to get it right . Any help ? 
I tried considering $a_n - a_{n+1}$ and I got the required proof but I don't feel as though this method is logical enough 
$a_n - a_{n+1} < 0$
$a_n < a_{n+1}$

Guide:


*

*Prove that $a_n>0$ if $a_1>0$.

*Compute $a_{n+1}-a_n$ in terms of $a_n$, your goal is to check that the expression is positive.

*check that the denominator is positive.

*Show that the numerator is positive using the property that $0<a_n<2$.

