- #1

Ratpigeon

- 56

- 0

## Homework Statement

Show that two metrics d_1,d_2 are equivalent iff for all epsilon>0, exists delta>0 such that

B_(d_1)(x,epsilon) [itex]\subset[/itex] B_(d_2)(x,delta) and vice versa (Where B_(d_1)(x, epsilon) is the open ball on the metric d_1 around x with radius epsilon.

## Homework Equations

pretty much what is in the first part

## The Attempt at a Solution

This is what I tried:

Let B_(d_1)(x,epsilon)=I, which is an open interval;

Assume

a [itex]\in[/itex] B_(d_1)(x,epsilon) and

a not [itex]\in[/itex] B_(d_2)(x,deltamax)

I'm trying to show this leads to a contradiction of deltamax being the largest delta that gives B_(d_2)(x,delta)[itex]\subset[/itex] I, but I'm not sure how.