An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$
Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$holds for all positive integers $m$ regardless of $a$
By induction $a_n=\frac{a}{a^n+1}$
$a_{n+1}=\frac{a}{\frac{a^2}{a_n}+1-a}=\frac{a}{a(a^n+1)+1-a}=\frac{a}{a^{n+1}+1}$
$a_1+a_1a_2+...+a_1...a_m=\frac{a}{a+1}+\frac{a^2}{(a+1)(a^2+1)}+...+\frac{a^m}{(a+1)...(a^m+1)}=1-\frac{1}{a+1}+ \frac{1}{a+1}-\frac{1}{(a+1)(a^2+1)}+...+\frac{1}{(a+1)...(a^{m-1}+1)}-\frac{1}{(a+1)...(a^m+1)}=1-\frac{1}{(a+1)...(a^m+1)} < 1$ for every $a$ snd $m$