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## Due September 4th

### Problem 1

Let $X$ be sequentially compact. Then every infinite subset of $X$ has a limit point. (Do not use the Bolzano-Weierstrass theorem.)

### Problem 2

Let $E$ be a compact subset of $\R$. Then it has a minimum and a maximum.

### Problem 3

Let $A$ be a bounded infinite subset of $\R^l$. Then it has a limit point.

### Problem 4

Let $(x_n)$ be a bounded sequence in $\R^l$. Then is has a convergent subsequence.

### Problem 5

Let $A$ be a nonempty set in the metric space $(X,d)$ and, for $\e>0$, define

$A_\e = \{x\in X: d(x,A) < \e\}.$

Then $A_\e$ is open in $X$.