Homework 1, Real Analysis

Due August 24

Problem 1

Two metrics $latex d_1, d_2$ on a space X are equivalent if there exist constants $latex c, C>0$ such that

$latex c d_1(x,y) \le d_2(x,y) \le C d_1(x,y)$

for all $latex x,y\in X$. Suppose $latex (X,d_1), (X,d_2)$ are equivalent.

1. They have the same convergent sequences.


2. They have the same Cauchy sequences.


3. $latex (X,d_1)$ is complete if and only if $latex (X,d_2)$ is complete.

Problem 2

Are $latex (C([0,1]), d_u)$ and $latex (C([0,1]), d_1)$ equivalent?

Problem 3

Are $latex (\R,|\cdot|)$ and $latex (\R,d_B)$ equivalent?

Problem 4

Suppose $latex (X,d_1)$ and $latex (X,d_2)$ have the same convergent sequences.

1. Are they equivalent?


2. Is one of complete if and only if the other is complete?