So, I was going through the AM-GM inequality and there is particular part I didn't quite understand. My question is: IF $c_1 <1$ and $c_k+1 > 1$ then how are they considering $c.c_2.c_3...c_k=1$ ?? I took it as being true assuming that $P(k)$ is true for $n=k$, but just because I'm denoting $c = c_1.c_k+1$ doesn't mean the value of $c_k+1$ is gone, moreover they said that $c_k+1 > 1$, so this could be anything then, $c_k+1$ can be $2, 10 , 100, 500$ so on...
In that case, how will this $c.c_2.c_3\ldots c_n= 1$ hold?? Please help!
Since $c_1\leq1$ and $c_{n+1}\geq1$, we obtain: $$(c_1-1)\left(c_{n+1}-1\right)\leq0$$ or $$c_1+c_{n+1}\geq1+c_1c_{n+1}.$$ Thus, by assumption of the induction (if $x_1x_2...x_n=1$ then $x_1+x_2+...+x_n\geq n$ for positives $x_i$)
we obtain: $$c_1+c_2+...+c_n+c_{n+1}=c_1+c_{n+1}+c_2+...+c_n\geq$$ $$\geq1+c_1c_{n+1}+c_2+...+c_n\geq1+\left(c_1c_{n+1}\right)c_2...c_n=1+n.$$