Let $\{a_{1,i}\}_{i=1}^k,\{a_{2,i}\}_{i=1}^k,\dots ,\{a_{n_,i}\}_{i=1}^k$ be real sequences. Does the following inequality hold

$$(\sum_{i=1}^k a_{1,i}^2)\cdot(\sum_{i=1}^k a_{2,i}^2)\cdots(\sum_{i=1}^k a_{n,i}^2)\geq (\sum_{i=1}^k a_{1,i}a_{2,i}\cdots a_{n,i})^2$$ for all $k,n \in \mathbb N$?

It can be easily seen that this is the Cauchy-Schwarz inequality when $n=2$.

The motivation for the problem actually comes from the Cauchy-Schwarz inequality. While solving a Cauchy-Schwarz inequality problem, this problem came to my mind. I don't know if this is already a proved theorem in mathematics (because I am a high school student and I don't know much about inequalities). But I didn't find this on internet (I searched on google). So, I assume the problem statement is false. And a proof (or disproof) is needed for that.

**My workings for $k=2$ and $n=3$:**

However, I tried to prove the problem statement for $k=2$ and $n=3$ (and I think I actually proved that!). Here is my workings to do that:

For $a,b,c,d,e,f$ real numbers, we have from Cauchy-Schwarz inequality (which is for $n=2$ and $k=2$),
$$(a^2+b^2)(c^2+d^2) \geq (ac+bd)^2$$
$$\implies (a^2+b^2)(c^2+d^2)(e^2+f^2) \geq (ac+bd)^2(e^2+f^2)$$
$$=(a^2c^2+2abcd+b^2d^2)(e^2+f^2)$$
$$=a^2c^2(e^2+f^2)+2abcd(e^2+f^2)+b^2d^2(e^2+f^2)$$
$$\geq a^2c^2e^2+2abcdef+b^2d^2f^2$$
$$=(ace+bdf)^2$$
as desired.

I hope my workings are correct. So, I have the following questions:

- Is the firstly stated problem statement true? If it is, how to prove that?
- If it is not true, are there some other values (like $k=2$ and $n=3$ as in the above) for which the statement is true?

Any help would be appreciated and please try to answer the questions so that a high school student can understand them (if it is not possible, then no problem).