prove that given x1x2...xn=1,
then x1+x2+x3+...+xn=n iff x1=x2=x3=...=xn=1
I tried to prove by induction. While the "if" direction is obvious, but I am kind of stuck in the "only if" direction.
Here's my take.
for n=2 given x1x2=1
only if: x1+x2=2, we have x1=2-x2 (2-x2)x2=1 x2=1 then x1 also =1
if: x1=x2=1 then, clearly x1+x2=2
for n=k assume x1x2...xk=1 then
x1+x2+x3+...+xk=k iff x1=x2=x3=...=xk=1
I am stuck here! please help