Here is my attempt at proving the problem below. 

$$\bigg|\dfrac{x^2y}{x^2+y^2}\bigg| \leq \bigg|\dfrac{x^2y}{2xy}\bigg| \text{ by the hint}$$ 

$$=\bigg|\dfrac{x}{2}\bigg| \leq |x| $$

Since we're using the delta epsillon definition to prove continuity, it should then follow that $|x|\leq|(x,y)|$. Have I missed anything or done something incorrectly here?

Thanks! 

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