Homework 1, Real Analysis

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?
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