How to prove an inequality for beginners

Question

I am a beginner in proofs and, unfortunately, I cannot wrap my mind around how to prove the simplest things, so I need a bit of help getting started. This is the proof that I am dealing with:

$\text{If }x< y< 0\text{, then }x^{2}> y^{2}\text{.}$

Thank you in advance.

Answer 1

We have that

$$x<y<0 \iff 0<-y<-x $$

and since $f(a)=a^2$ is strictly increasing for $a>0$

$$0<-y<-x \implies 0<(-y)^2<(-x)^2$$

that is

$$x^2>y^2>0$$

Answer 2

$x^2>y^2\Rightarrow|x|>|y|\Rightarrow y>x$ for $x,y<0.$

Answer 3

If $x < y < 0$ then $|x| > |y|$ so $$x^2 = |x|^2 > |y|^2 = y^2$$

Answer 4

Multiplying an inequality by a negative number reverses the inequality.

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We have $x<y$ and multiplying it with negative number $x$ we get $x^2>xy$. Again multiplying the same starting inequality by negative number $y$ we get $xy>y^2$. Now using $x^2>xy$ and $xy>y^2$ we arrive at $x^2>y^2$.