Simple two variable am-gm inequality Given $x,y \in \Bbb{R}$, show that:$$x^2+y^2+1\ge xy+y+x $$
I tried using the fact that $x^2+y^2 \ge 2xy$ But then I'm not sure how to go on, Also tried factoring but didn't help much, also tried substituting $\frac{x^2+y^2}{2}$ instead of $xy$ but that gave me the same result of the first substitution, i.e. $xy+1\ge x+y$
This inequality seems very easy, I'm feeling dumb for not having solved it yet

We need to prove that
$$y^2-(x+1)y+x^2-x+1\geq0,$$ for which it's enough to prove that
$$(x+1)^2-4(x^2-x+1)\leq0$$ or $$(x-1)^2\geq0.$$
