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Let $(X,d)$ be a metric space. Let $A \subset X$ and $c \in X$. $c$ is called an accumulation point of $A$ if for every $\delta > 0$ there exists $a \in A$ such that $0 < d(a,c) < \delta$. The set of accumulation points of a set $A$ is denoted by $A'$. For example if $\Bbb R$ is endowed with the usual metric, then $\Bbb Q' = \Bbb R$, $\Bbb N'= \varnothing$, $(a,b)' = [a,b]$. Let $A$, $B$, $C$, $D$ be subsets of $X$. Prove the following statements.

(a) $C \subset D \Longrightarrow C' \subset D'$

(b) $(A \cup B)' = A' \cup B'$

(c) $\overline{A} = A \cup A'$

(d) $A$ is closed if and only if $A' \subset A$

(e) If $B$ is finite, then $B' = \varnothing$

(f) If $B$ is a finite subset of $A$, then $A' = (A \setminus B)'$. Note that $A = B \cup (A \setminus B)$

(g) Let $(x_n)$ be a sequence in $X$. If $A = \{x_n : n \in \Bbb N\}$ and $a \in A'$, then $(x_n)$ has a subsequence converging to $a$. (Use induction and (f) to construct a strictly increasing sequence $(k_n)$ of integers such that $0)

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    What do you think? What have you tried? How can you apply the definitions you need to tackle these problems?2012-12-02
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    Actually, I am a physics student and I joined a class in Math dept. because I really want to learn Functional Analysis. Because of that I don't have basic information, I could not find a solution to this question myself. I will be appreciated if you can help me.2012-12-02

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