Homework 1, Real Analysis

Due date: August 11

Problem 1

  1. Let d_M be the function on \R^n given by d_M(x,y) = \max\{ |x^1 - y^1|, \ldots, |x^n - y^n|\}. Then d_M is a metric.
  2. Let d_T be the function on \R^n given by d_T(x,y) = |x^1 - y^1| + \ldots + |x^n - y^n|. Then d_T is a metric.

Problem 2

For a metric space (X,d), define d_B(x,y) = \dfrac{d(x,y)}{1 + d(x,y)}.

  1. Are the metrics d, d_B equivalent?
  2. Do they have the same convergent sequences?

Problem 3

A discrete metric space (X,d) is complete.

Problem 4

Give an example of a pair of metric spaces with the same convergent sequences, but such that one is complete and the other is not.

Problem 5

Let x_n, y_n sequences in the metric space (X,d) such that

d(x_n, y_n) \to 0.

Then x_n converges if and only if y_n does.

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