Prove With Three Real Numbers Prove That We Can Pick Two And Their Product is Non Negative

Proof by contradiction.

We want to prove that given any triple of numbers $a,b,c\in\mathbb{R}$ we can extract two such that their product is non negative.

Suppose that there exists a triple $(x,y,z)$ such that any of the three possible products $$xy,\;xz,\;yz$$ is negative. So their product should be negative as well. But $$(xy)(xz)(yz)=(xyz)^2$$ Contradiction.

Thanks to @Servaes for the collaboration.

Edit.

This property can be easily generalized to any set $S \subseteq\mathbb{R}$ containing more than two elements.

We want to prove that from any of those subsets it can be chosen a pair of elements $(x,y)\in S\times S$ such that $xy\ge 0$. Indeed suppose all pairs $(a,b)\in S\times S$ are such that $ab<0$ we could chose $(p,q)$ and $(p,r)$ such that $pq<0,\;pr<0$. So $(pq)(pr)>0$ which means $p^2(qr)>0$ and as $p^2>0$ we would have $qr>0$ in contradiction with the assumpt that all pair gave a negative product.

I'll assume that here that $a$, $b$ and $c$ must be nonzero, as otherwise the statement to prove is false.

You've treated two cases, but there are many more cases. As each of $a$, $b$ and $c$ is either positive or negative (you seem to exclude the possibility that the variables are $0$), there are $2^3=8$ cases to consider. Of course this is cumbersome, and it is not the way to go about things.

One way to reduce the number of cases, is to consider the number of negative variables among $a$, $b$ and $c$. This leaves just $4$ cases to consider. Another approach is the following:

The product of two nonzero real numbers is positive if and only if their signs agree. By the pigeonhole principle, given three nonzero real numbers, there must be two with the same sign. So you can pick two of them so that their product is positive.

Edit: Another way to keep the statement to prove from being false is to interpret positive as meaning nonnegative. Then the numbers $a$, $b$ and $c$ may equal $0$, but of course in this case the product is also $0$.

Have you ever heard the puzzle if you have a drawer full of black and blue socks and you are blindfolded and need to pull out a pair of matching socks, how many do you need to pull?

Answer: $3$.

If you only have $2$ categories for things to be (blue or blue) and you pull out $3$ off them, as you can't have all $3$ items be different because you have more items then categories so at least $2$ must be in the same category. And if you want to spell it out; you can have all three socks black (and you have more than two matching socks); you can have two black and one blue (and you have a pair of matching black socks); you can have one black and two blue (and you have one pair of matching blue socks); or you can have three blue socks (and you have more than one pair of matching socks).

This is the exact same thing. All real numbers are either 1) negative or 2) not negative. So if you draw three real numbers at random the at least two of the them will be the same type.

If you have at least two negatives, then their product is the product of two negatives. That is positive, and so the product is non negatives.

If you don't have at least two negatives, then you have at most one negative and you have at least two numbers that are not negative. If you multiply two non-negative numbers together you can not get a negative number. (Either they are both positive; or one is zero; or both are zero-- in any case the product can not be negative.)