Proving the Inequality kx > ky for x < y and k < 0

• rhule009
In summary, a scientific theory can be proven through extensive testing and evidence, using the scientific method. A hypothesis can also be proven or disproven through experimentation. However, theories can be proven wrong if new evidence is discovered. The scientific method is the most reliable way to prove something in a scientific context.
rhule009

Homework Statement

show that if x and y are real numbers such that x<y, then for any real number k<0, kx>ky

The Attempt at a Solution

I think a proof by contradiction should work.

First, let's let k = -b, with b > 0, for simplicity.

Now, given x < y, we know that 0 < y - x.

Then, assume that -bx < -by.

It follows:

-bx - (-by) < 0
-b(x-y) <0

Since b is non-zero, we may divide it out (I suppose this depends on the fact that bx < by implies x < y for b > 0, so I'm assuming this has already been proved).

Then,

-(x-y) = y - x < 0

But, our original statement is that y - x > 0, so we have a contradiction, which means that -bx > -by.

(I think this might also require the fact that we know -bx = -by is true only when x = y, which is again a violation of x < y, otherwise I don't think this proof alone rules out that possibility).

Last edited:
You don't need an argument by contradiction if you accept the "rule of signs" for multiplying numbers:

+ times + is +
+ times - is -
- times + is -
- times - is +

x < y
x-y < 0
If k < 0 then k(x-y) > 0 (minus times minus = plus)
kx - ky > 0
kx > ky

And where do you pull your "rule of signs" from?
It is a higher-order statement that the one to be proven.

arildno said:
And where do you pull your "rule of signs" from?
It is a higher-order statement that the one to be proven.

I learned it in elementary school, about 50 years ago.

I assumed this forum it was intended to help people, not to show off ones own knowledge and/or have p*ssing competitions. Apologies if I was wrong about that.

If you want to post a proof starting from Peano's axioms and a definition of the real numbers using Dedekind cuts, feel free. But I doubt if the OP would benefit much from reading it.

This formum is not about getting huffy is someone objects to what you have done. The real problem is that the original poster did not show any work and so we have no idea what "basis" he is starting from- and so neither you nor I have any idea what the OP would benefit from! Certainly, if you are allowed to use the fract that "positive times positive is positive" the "proof" is trivial. That is why I would suspect that that is not what the OP wanted.

The rule of signs follows directly from the axioms for an ordered field.

Axiom 1: if a > b, then for any c, a+c > b+c

As a special case, take b = 0 and c = -a: then Axiom 1 becomes

If a > 0 then -a < 0
Similarly, if a < 0 then -a > 0

Axiom 2: if a > 0 and b > 0 then ab > 0

So to prove one case of the rule of signs:

if a < 0 and b > 0, then -a > 0 (axiom 1)
(-a)(b) > 0 (axiom 2)
-(ab) > 0 (Field axioms of arithmetic)
ab < 0 (axiom 1)

I don't understand Arildno's comment that the rule of signs is "a higher order statement than the one to be proven".

True. If someone were to ask me how to prove "if x< y then for any k< 0, ky< kx", I would probably start with the axiom: "if a> 0 and b> 0 then ab> 0", or, equivalently "if a< b and c> 0 then ac> bc". Of course, we still don't know what rhule009 has to work with!

1. Can someone prove a scientific theory?

Yes, a scientific theory can be proven. In science, a theory is an explanation or model that has been extensively tested and supported by evidence. It is the highest level of scientific understanding and can be proven through repeated experimentation and observation.

2. How do scientists prove their theories?

Scientists prove their theories through the scientific method. This involves making observations, forming a hypothesis, conducting experiments, analyzing data, and drawing conclusions. The results of these experiments can either support or disprove a theory.

3. Is it possible to prove a hypothesis?

Yes, it is possible to prove a hypothesis. A hypothesis is a proposed explanation for a phenomenon that can be tested through experimentation. If the results of the experiment support the hypothesis, it can be considered proven. However, a hypothesis can also be disproven if the results do not support it.

4. Can a scientific theory ever be proven wrong?

Yes, a scientific theory can be proven wrong if new evidence or data is discovered that contradicts it. In science, theories are constantly being tested and revised as new information becomes available. This is a normal and necessary part of the scientific process.

5. Is it possible to prove something without using the scientific method?

No, the scientific method is the most reliable way to prove something in a scientific context. It involves rigorous experimentation and data analysis to ensure that the results are accurate and can be replicated by others. Other methods of proof may be used in different fields, but in science, the scientific method is the standard for proving theories and hypotheses.

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