multivariable calculus - Indeterminate two-dimensional limit

I'm pretty sure that

\begin{equation}
\lim_{(x,y) \rightarrow (0,0)} \frac{x^4y}{x^2 + y^2} = 0,
\end{equation}

but I'm having some trouble proving it.

The only technique I'm aware of that can be used to show indeterminate limits of $\geq 2$ variables exist is the Squeeze Theorem. I've tried applying it here (by assuming $|y| < 1$ and bounding the quantity of interest by $\pm\frac{x^4y}{x^2 + y^2}$), but I didn't get anywhere.

Any help is appreciated.

Answer

Since $|xy|\leq \frac{x^2+y^2}{2}$ by the GM-QM inequality, you simply have:
$$\left|\frac{x^4 y}{x^2+y^2}\right|\leq \frac{1}{2}|x|^3.$$