MHB Can Sum to Product Inequalities Hold for Non-Negative Reals?

AI Thread Summary
The discussion centers on proving that for non-negative reals $\alpha_i$, if the sum $\alpha_1 + \alpha_2 + ... + \alpha_n \leq \frac{1}{2}$, then the product $(1 - \alpha_1)(1 - \alpha_2)...(1 - \alpha_n) \geq \frac{1}{2}$. Participants express appreciation for a clever solution provided by June29, indicating it may have addressed the proof effectively. There is a repeated request for an inductive proof of the statement, highlighting interest in different methods of validation. Overall, the thread emphasizes the exploration of inequalities involving non-negative reals and their implications.
lfdahl
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Given non-negative reals, $\alpha_i$, where $i = 1,2,...,n.$

Prove, that

$\alpha_1+\alpha_2+...+\alpha_n \leq \frac{1}{2}$ $\Rightarrow$ $(1-\alpha_1)(1-\alpha_2)...(1-\alpha_n) \geq \frac{1}{2}.$
 
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Immediate consequence of Weiertrass inequality:

$$\prod_{1 \le k \le n}(1-a_k) \ge 1-\sum_{1 \le k \le n}a_k \ge 1-\frac{1}{2} = \frac{1}{2}.$$

There's an elegant proof of Weierstrass inequality.
 
June29 said:
Immediate consequence of Weiertrass inequality:

$$\prod_{1 \le k \le n}(1-a_k) \ge 1-\sum_{1 \le k \le n}a_k \ge 1-\frac{1}{2} = \frac{1}{2}.$$

There's an elegant proof of Weierstrass inequality.
Thankyou for a clever solution, June29, and for your participation!(Cool)
 
Can anyone prove the above statement by induction? (Wave)
 
lfdahl said:
Can anyone prove the above statement by induction? (Wave)

It's obviously true for $n=1$ since we have $ {\alpha}_1 \leqslant \frac{1}{2} = 1-\frac{1}{2} \implies 1-\alpha_1 \geqslant \frac{1}{2}. $

Now, suppose it's true for $n = k \in \mathbb{N}$. We shall prove that it's true for $n=k+1$. $\displaystyle \begin{aligned} \frac{1}{2} & \leqslant 1- \sum_{1 \leqslant j \leqslant k+1}\alpha_j = 1-\alpha_{k+1}-\sum_{1 \leqslant j \leqslant k}\alpha_j \leqslant 1-\alpha_{k+1}-\sum_{1 \leqslant j \leqslant k}\alpha_j+a_{k+1}\sum_{1 \leqslant j \leqslant k} \alpha_j \\& =\left(1-\alpha_{k+1}\right)\left(1-\sum_{1 \leqslant j \leqslant k} \alpha_j\right)
\leqslant \left(1-\alpha_{k+1}\right) \prod_{1 \leqslant j \leqslant k} \left(1-\alpha_j \right) = \prod_{1 \leqslant j \leqslant k+1} \left(1-\alpha_j \right) \end{aligned} $

So it's true for $n=k+1$. Since it's true for $n=1, k+1$, it's true for all $n\in\mathbb{N}$.
 
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June29 said:
It's obviously true for $n=1$ since we have $ {\alpha}_1 \leqslant \frac{1}{2} = 1-\frac{1}{2} \implies 1-\alpha_1 \geqslant \frac{1}{2}. $

Now, suppose it's true for $n = k \in \mathbb{N}$. We shall prove that it's true for $n=k+1$. $\displaystyle \begin{aligned} \frac{1}{2} & \leqslant 1- \sum_{1 \leqslant j \leqslant k+1}\alpha_j = 1-\alpha_{k+1}-\sum_{1 \leqslant j \leqslant k}\alpha_j \leqslant 1-\alpha_{k+1}-\sum_{1 \leqslant j \leqslant k}\alpha_j+a_{k+1}\sum_{1 \leqslant j \leqslant k} \alpha_j \\& =\left(1-\alpha_{k+1}\right)\left(1-\sum_{1 \leqslant j \leqslant k} \alpha_j\right)
\leqslant \left(1-\alpha_{k+1}\right) \prod_{1 \leqslant j \leqslant k} \left(1-\alpha_j \right) = \prod_{1 \leqslant j \leqslant k+1} \left(1-\alpha_j \right) \end{aligned} $

So it's true for $n=k+1$. Since it's true for $n=1, k+1$, it's true for all $n\in\mathbb{N}$.

A nice solution, June29! Thankyou for your participation!

Please remember to hide your solution in SP tags. Other forum users might try to solve the challenge preferably without knowing your solution. Thankyou in advance!
 
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