# Continuity Proof

1. Mar 14, 2013

### gajohnson

1. The problem statement, all variables and given/known data

The problem statement and proof can be found here. The proof continues after this, but I only have a question about the beginning of the proof.

2. Relevant equations

NA

3. The attempt at a solution

My question is simply this:

Every proof I find for this problem starts assuming the inequality $0≤(x-y^2)^2$ and I cannot figure out why. I'm sure this is a simple matter, but any explanation would be appreciated.

Thanks!

2. Mar 14, 2013

### jbunniii

The square of a real number can't be negative, right?

3. Mar 14, 2013

### gajohnson

Yes, indeed, I see that it works. But was this first step just conjured out of experience and a keen eye, or did something in the problem suggest it specifically?

Thanks!

4. Mar 14, 2013

### jbunniii

Probably experience and a keen eye. If you play around with inequalities often enough, you start to recognize things like the fact that the numerator of
$$\frac{xy^2}{x^2 + y^4}$$
can be used to "complete the square" in the denominator, i.e.
$$x^2 + y^4 + 2xy^2 = (x + y^2)^2$$
and
$$x^2 + y^4 - 2xy^2 = (x - y^2)^2$$
If you don't notice this trick, there are usually other ways to find a bound.