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.

It indeed does fly. If you multiply both sides of an inequality by a positive quantity, the inequality is preserved.   
