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## Homework Statement

Define the notion of an open set in Rn.

Prove that the set {(x

^{4})(y

^{4})(z) | x > 0; y > 0; z < 0 } is open in R3.

## Homework Equations

Definition of an open set

B

_{r}(p) = { x is an element in R

^{n}|x - p| < r} c U

## The Attempt at a Solution

Well we first pick a point p in U, and we want to find and r > 0 s.t. B

_{r}(p) is in U.

Choose r = min(|x

^{4}||y

^{4}||z|)

Then q= (x`, y`, z`) element of B

_{r}(p)

(x` - x

^{4})

^{2}+ (y` - y

^{4})

^{2}+ (z` - z)

^{2}< r

^{2}

And from here I have to show x` > 0, y` > 0, z` < 0

x and y will follow the same way procedurally; from the inequality above, we know

(x` - x

^{4})

^{2}< r

^{2}

|x` - x

^{4}| < r

Here is where I get stuck; with exponents I am finding this more difficult to rearrange, am I supposed to use this inequality;?

x

^{2}+ y

^{2}>= 2xy, by comparing both x and y to r?

Any help or insight would be great thanks, proving open sets have been giving me a lot of difficulty.