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Due August 24

Problem 1

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

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

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

1. They have the same convergent sequences.

2. They have the same Cauchy sequences.

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

Problem 2

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

Problem 3

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

Problem 4

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

1. Are they equivalent?

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