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## 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.