How do you prove that x^2+y^2-2xy+x-y+1 is greater than 0?
You can't.
$\displaystyle \displaystyle \begin{align*} x^2 - 2\,x\,y + y^2 + x - y + 1 &= \left( x - y \right)^2 + x - y + 1 \end{align*}$
All that you can say for sure is that $\displaystyle \displaystyle \begin{align*} \left( x - y \right)^2 \end{align*}$ is nonnegative. The entire quantity will only be positive if $\displaystyle \displaystyle \begin{align*} x - y + 1 > 0 \end{align*} $, which you are not guaranteed.