I'm trying to solve the below problem from a University of Oxford problem sheet.
Given $n$ positive integers $x_1, x_2, \ldots, x_n$ such that $x_1 + x_2 + \ldots + x_n \leq \frac{1}{3}$, prove by induction that $$ (1 - x_1) (1 - x_2) \cdots (1 - x_n) \geq \frac{2}{3}. $$
I'm not completely sure whether the way this is written implies that $n$ needs to be at least $2$. Otherwise, I'd be inclined to write $x_1, \ldots, x_n$ to emphasize that $n = 1$ is possible. Regardless, here's my attempt. The problem gives a hint to consider $x_1, \ldots, x_{n-1}, x_n + x_{n+1}$ for the inductive step. I'm not sure how to use this or where the sum of the $n$th and $(n+1)$st terms comes from. That seems to me to sacrifice generality.
Let $n = 1$. Then $x_1 \leq \frac{1}{3}$, so $1 - x_1 \geq 1 - \frac{1}{3} = \frac{2}{3}$. Suppose that this statement holds for $k$, and suppose that $x_1, \ldots, x_k + x_{k+1} \leq \frac{1}{3}$. As the $x_i$ are positive, we know that $$ \sum\limits_{i=1}^{k} x_i \leq \sum\limits_{i=1}^{k+1} x_i \leq \frac{1}{3}. $$ The induction hypothesis then implies that $$ \prod\limits_{i=1}^k (1 - x_i) \geq \frac{2}{3}. $$
I'm stuck at this point. I'm not sure whether I should write out an $n = 2$ base case to make the transition from $k$ to $k+1$. I think I need to show that $1 - x_{k+1} > 0$. That means that $x_{k+1} < 1$. If the sum $\sum\limits_{i=1}^k x_i \leq \frac{1}{3}$ and $x_{k+1} \geq 1$, then $\sum\limits_{i=1}^{k+1} x_i > \frac{1}{3}$, which is a contradiction. Is that on the right track?
For the inductive step, we want to show that if $x_1,\dots,x_k,x_{k+1}$ are positive numbers such that $x_1+\dots+x_k+x_{k+1}\le 1/3$, then $(1-x_1)\cdot\dots\cdot(1-x_{k+1})\ge 2/3$.
What we know by inductive hypothesis is that for any positive numbers $y_1,\dots,y_k$ such that $y_1+\dots+y_k\le 1/3$, we have $(1-y_1)\cdot\dots\cdot(1-y_k)\ge2/3$. The hint is suggesting you consider the particular assignment $$ y_1 = x_1,\dots,y_k=x_k+x_{k+1}. $$ When you apply the inductive hypothesis to these particular $y_1,\dots,y_k$, what do you get?
Note that it doesn't have to be just this assignment which would work. Another assignment which would work is $$ y_1 = x_1 + x_2, y_2 = x_3, \dots, y_k = x_{k+1}. $$ The point is to make an assignment that connects the expression $x_1+\dots+x_{k+1}$ with $y_1+\dots+y_k$ for some numbers $y_1,\dots,y_k$, so that we can make use of our inductive assumption regarding a sum of $k$ positive numbers.