Prove this inequality using induction

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Homework Statement

Given n positive numbers x1, x2, . . . , xn such that x1 + x2 + · · · + xn <= 1/3, prove by
induction that
(1 − x1)(1 − x2) × · · · × (1 − xn) >= 2/3

Relevant Equations

Principle of Induction, proof by induction, base case, inductive step

Been stuck on this one for a while now.

Base case is easy, n=1, we have x <=1/3, so trivially 1-x>= 2/3 and we are done.

The issue is with the inductive step, I don't know how to use the hint, infact I am struggling to understand what is meant by the hint.

Any help (or a full solution) would be greatly appreciated.

 

[PeroK]

The issue is with the inductive step, I don't know how to use the hint, infact I am struggling to understand what is meant by the hint.

The hint means that when you have $n + 1$ numbers, you combine the last two numbers by adding them, and then you only have $n$ numbers.

 

[Orodruin]

To make that a little more explicit: Consider $x_1 + x_2 = y \leq 1/3$. Then by the $n=1$ case, $(1-y) \geq 2/3$. Therefore $(1-x_1)(1-x_2) = 1 - y + x_1 x_2 \geq 1-y \geq 2/3$, which proves the relation for $n=2$. Now generalize this to an induction step.