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## Homework Statement

If A is a discrete subset of the reals,

prove that

[tex]A'=cl_x A \backslash A[/tex]

is a closed set.

## Homework Equations

A' = the derived set of A

x is a derived pt of A if [tex]U \cap (A \backslash \{x\}) \neq \emptyset[/tex] for every open U such that x is in U.

Thrm1. A is closed iff A=cl(A)

Thrm2. cl(emptyset)=emptyset

## The Attempt at a Solution

"Proof". Using Thrm1 with A' we can see that A' is closed iff A'=cl(A'). Since A is a discrete subset of the Reals we know that the set consists of isolated pts. Since there are no derived pts in A, then A'= emptyset. Using Thrm2 we know that cl(emptyset)=emptyset. Therefore A'=cl(A'). And thus A' is closed.

Does this proof work?