How did they come up with the inequality for the multivariable function

Question

I am trying to figure out how in my answers sheet they got the approximation of the function $$g(x,y)=\frac{y}{1+x^2+y^2}$$ to $$\mid{g(x,y)} \mid \leq \frac{\sqrt{x^2+y^2}}{1+x^2+y^2} $$

Is it simply because $\sqrt{x^2+y^2}$ is a positive number and thus y is smaller than something added to it?

Answer 1

It's because$$\lvert y\rvert=\sqrt{y^2}\leqslant\sqrt{x^2+y^2}.$$

Answer 2

Since $$|g(x,y)|=\frac{|y|}{1+x^2+y^2}$$ and $$|y|\le\sqrt{x^2+y^2}$$ you will get your inequality.