derivatives of multi variable fuction

Question

I have just learned about multivariable functions and I can't understand: $$\frac{\partial}{\partial x} \frac{\partial}{\partial y}f(x,y) =\frac{\partial}{\partial y}\frac{\partial}{\partial x}f(x,y)$$ I understand how you take the derivative of these functions, and the prof explained why they are equal but I didn't understand it too well. He wrote out the full $\frac{\partial^2}{\partial x \partial y}$ derivative using Newton's method of taking derivatives, rearranged it in some kind of way and got the other derivative $\frac{\partial^2}{\partial y \partial x}$ but at the end he said that it doesn't always work and when some limit converges the derivative exists and at that point I lost him. Do you have a good explanation for why it works?