- #1

julypraise

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## Homework Statement

I've proved that if [itex] B = \bigcup_{i=1}^{\infty} A_{i} [/itex] then [itex] \overline{B} = \bigcup_{i=1}^{\infty} \overline{A_{i}} [/itex] but it should not be right. So could you find errors on my reasoning?

## Homework Equations

## The Attempt at a Solution

Observe [tex] x \in \overline{B} [/tex]

iff for every [tex] \epsilon>0 \quad B(x;\epsilon) \cap B \neq \emptyset [/tex]

iff [tex] B(x;\epsilon) \cap \bigcup_{i=1}^{\infty} A_{i} \neq \emptyset [/tex]

iff [tex] B(x;\epsilon) \cap A_{i_{0}} \neq \emptyset [/tex] for some [itex] i_{0} \in \mathbb{Z}^{+}[/itex]

iff [tex] x \in \overline{A_{i_{0}}} [/tex]

iff [tex] x \in \bigcup_{i=1}^{\infty} \overline{A_{i}} [/tex]