# Math Help - continuous metric spaces

1. ## continuous metric spaces

Show that

1. the projection map p : R^2 → R given by p(x; y) := x is continuous.
.

My proofs:

We want to show that $\forall$ $\epsilon$ > 0, $\exists$ $\delta$>0 such that:

d(x,y)< $\delta$ $\Rightarrow$ d(f(x),f(y)) < $\epsilon$

(x1,y1) (x2,y2) are points in R^2

1) d(x1,x2) < $\epsilon$
|x1-x2| < $\epsilon$

We can define $\delta$ = the square root of ( $\epsilon$^2 - (y2-y1)^2)

Since there exists a delta for any epsilon we choose, implies that p is continuous.

Is this correct...

2. Looks quite correct to me.