Multivariable epsilon-delta proof

[nafisanazlee]

Homework Statement

$$\lim\limits_{(x,y) \to (1,2)} (5x^3 - x^2y^2)$$

Relevant Equations

For every $$\epsilon > 0$$, we must find $$\delta > 0$$ such that whenever
$$
\sqrt{(x - 1)^2 + (y - 2)^2} < \delta
$$
it follows that
$$
| (5x^3 - x^2y^2) - 1 | < \epsilon.
$$

I have provided my solution here. I just want to be sure if it's correct or not.

 

[FactChecker]

There is a lot going on in obtaining the second-to-last step (obtaining the line of the strict inequality).
It seems like there could be a mixture of signs of terms that need a more detailed explanation.

 

[FactChecker]

Do you have any theorems about sums and products of continuous functions that you can use?

 

[nafisanazlee]

I used the triangular inequality.

 

[FactChecker]

I used the triangular inequality.

I think we are not talking about the same line. Are you sure about the logic giving the line with the strict inequality? It's such a combination of signs and substitutions that I can not follow it. Maybe if you applied the triangle inequality before that line to break up the terms into separate absolute values it would be easier to follow.
ADDED: I'm not saying it's wrong. I just can't follow it and would like something simpler.