I am studying for a midterm, and I came across this proof.
Use mathematical induction(weak) to show that for all integers n $\geq$ 2 , if x$_{1}$, x$_{2}$, ... x$_{n}$ are strictly between 0 and 1 then the following holds true.
(1-x$_{1}$)(1-x$_{2}$)...(1-x$_{n}$) $\geq$ 1-x$_{1}$-x$_{2}$-...-x$_{n}$
This is a screen shot of the parts of the the part I am having trouble understanding.
My question is, why in the last step could it be assumed that x$_{n}$ is less than x$_{n}$(x$_{1}$ + x$_{2}$ +...+ x$_{n-1}$)? The rest of the proof seems self explanatory.