Prove that positive $x,y$ satisfy $\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$. 
Prove that positive $x,y$ satisfy $$\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$$

My teacher claims this lemma is often useful. How to prove it?
I've tried using $a^2+b^2\ge 2ab$ and $a^2+b^2\ge \frac{(a+b)^2}{2}$ and $\frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+\ldots+\frac{a_n^2}{b_n}\ge \frac{(a_1+a_2+\ldots+a_n)^2}{b_1+b_2+\ldots+b_n}$.

Not particularly elegant, but if you multiply everything out, you get
$$x^3y+xy^3+1\ge x^2y^2+2xy$$
And that's just $2$ simple AM-GM inequalities
$$x^3y+xy^3\ge2x^2y^2$$
$$x^2y^2+1\ge2xy$$
