Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$ Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$
Here is what I would do if this were asked on a test and I was told to "justify" the answer.
Let $x \in \mathbb{R}$
Assume $x$ is greater than $1$.
Then $x * x > x$ , since $x$ is greater than $1$. 
Therefore $x^2 > x \square$
Not sure how that will fly with the grader or this community. What I would like to know is if I have correctly shown that the statement is true? How would you have justified that this is a true statement? It is these really obviously true or false statements that I have trouble proving.

If $x > 1$, then $x > 0$ and $x-1 > 0$
so $x(x-1) > 0$
or $x^2-x > 0$
or $x^2 > x$.
An interesting variant on this
is to show that
if $x > 0$
then, unless $x = 1$,
$x^2$ is further from $1$ than $x$ is
and $\sqrt{x}$ is closer to $1$
than $x$ is.
For extra credit, 
do this without separating into the cases
$x < 1$ and $x > 1$.
