Basic question about inequality let $x>1$
Obviously $x^2>x$
Then $x^2>x>1$
Taking $x^2>1$, we can assert that this holds true for all the values for $x>1$ and $x<-1$
But if I take $-5$ such that $x<-1$, then $x^2>x$ holds but $x>1$ doesn't. Why is it so? Doesn't $x^2>x>1$ mean that all of the three must be true?

Yes, $x^2>x>1$ is really shorthand for "$x^2>x$ and at the same time $x>1$". So it doesn't hold for $x = -5$.
You start with "let $x>1$". This is not true for $x = -5$. You do get $x^2 > 1$ in this case, but your argument as written won't get you there. You need a different argument.
