Looks quite correct to me.
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 > 0, >0 such that:
d(x,y)< d(f(x),f(y)) <
(x1,y1) (x2,y2) are points in R^2
1) d(x1,x2) <
|x1-x2| <
We can define = the square root of ( ^2 - (y2-y1)^2)
Since there exists a delta for any epsilon we choose, implies that p is continuous.
Is this correct...