Answer to Question #308333 in Real Analysis for Pankaj

Question

Which of the following statements are true and which are false? Justify your answers with

a short proof or a counter-example:

If x and y are real numbers such that x < y, then x²<y²

Answer 1

Solution

Given that

We need to check the statement that, if $x$ and $y$ are real numbers such that $x < y$, then $x²<y²$

let us consider $-2<-1$

Then

$(-2)^2>(-1)^2\\ 4>1$

Hence the given statement is False, because for all real $x$ and $y$, such that $x < y$, it is not always true $x²<y²$. It is true only when both $x$ and $y$ are positive real numbers.