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Homework 1, Real Analysis

Due date: August 11

Problem 1

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

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

Problem 2

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

  1. Are the metrics $latex d, d_B$ equivalent?

  2. Do they have the same convergent sequences?

Problem 3

A discrete metric space $latex (X,d)$ 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 $latex x_n, y_n$ sequences in the metric space $latex (X,d)$ such that

$latex d(x_n, y_n) \to 0$.

Then $latex x_n$ converges if and only if $latex y_n$ does.