- #1

Mogarrr

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## Homework Statement

Let [itex] A_1, A_2, A_2,...[/itex] be subsets of a metric space.

(a) If [itex]B_{n}=\bigcup_{i=1}^{n}A_{i} [/itex], prove that [itex] \bar{B_{n}}= \bigcup_{i=1}^{n}\bar{A_{i}} [/itex] for n=1,2,3,...

(b) If

**[itex]B=\bigcup_{i=1}^{\infty}A_{i} [/itex], prove that**[/B]

**[itex] \bar{B} \subset \bigcup_{i=1}^{\infty}\bar{A_{i}} [/itex],**

Show by example this inclusion is proper.Show by example this inclusion is proper.

## Homework Equations

If E is a subset of a metric space, then the closure of E,

[itex] \bar{E}=E \bigcup \acute{E} [/itex], where [itex] \acute{E} [/itex] is the set of all limit points of E.

For a subset E of a metric space X, a point [itex] p \in X [/itex] is a limit point if and only if

[itex] \forall \epsilon > 0, \exists q \in B(p;\epsilon) \bigcap E : q \neq p [/itex]

## The Attempt at a Solution

**Now I think I've got part (a) and (b) nailed down, but I'm having trouble thinking of an example for a proper subset. Here's what I have thus far...**

**Suppose [itex]x \in \bar{B_{n}} [/itex], then [itex] x \in B_{n} [/itex] or x is a limit point for [itex] B_{n} [/itex].**

If[itex] \forall \epsilon > 0, \exists y \in B(x;\epsilon) \bigcap B_{n}: y \neq x [/itex], so [itex] \forall \epsilon > 0, \exists y \in B(x;\epsilon) \bigcap (\bigcup_{i=1}^{n} A_{i}): y \neq x [/itex], then

If

**[itex]x \in {B_{n}} [/itex], then****[itex]x \in \bigcup_{i=1}^{n} A_{i} [/itex], then [itex] \exists i, x \in A_{i} [/itex], so [itex] \exists i, x \in \bar{A_{i}} [/itex].**

If x is a limit point, thenIf x is a limit point, then

[itex] \exists i,\forall \epsilon > 0, \exists y \in B(x;\epsilon) \bigcap A_{i}: y \neq x [/itex], then

[itex] \exists i, x\in \bar{A_{i}}[/itex], so [itex] x \in \bigcup_{i=1}^{n}\bar{A_{i}}[/itex]

Suppose [itex] x \in \bigcup_{i=1}^{n}\bar{A_{i}}[/itex], then [itex] \exists i, x \in A_{j}[/itex] or for some i x is a limit point.

If [itex] \exists i, x \in A_{i}[/itex], then [itex] x \in \bigcup_{i=1}^{n}A_{i}=B_{n}[/itex].

If x is a limit point for some A sub i, then x is a limit point for [itex] \bigcup A_{i}=B_{n} [/itex].

For part (b), I did the same thing, but replaced [itex] B_{n} [/itex] with [itex] B [/itex] and in the union I replaced n with infinity.

Now, I'm having some difficulty coming up with an example that would show B is a proper subset of the union of an infinite number of A sub i's.

Any hints or corrections would be much appreciated.