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Tarea 4, Análisis real

Due September 4th

Problem 1

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

Problem 2

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

Problem 3

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

Problem 4

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

Problem 5

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

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

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