1. ## Derived Set Proof

Hello, I am currently studying for my analysis midterm, and I can't seem to figure out how to solve the following:

Let S,TRn. Show that (SUT)' = S'UT'.

Where S' and T' represent derived sets.

Any help with this problem would be greatly appreciated!

2. Originally Posted by Majialin
Hello, I am currently studying for my analysis midterm, and I can't seem to figure out how to solve the following:

Let S,TRn. Show that (SUT)' = S'UT'.

Where S' and T' represent derived sets.

Any help with this problem would be greatly appreciated!
x is an accumulation point of (SUT)' iff there exists a sequence of distinct points a_n of SUT with |x - a_n| < 1/n. This is true iff either infinitely many of the points belong to S or infinitely many belong to T (or both), which is true iff either x belongs to S' or x belongs to T'. Hence (SUT)' = S'UT'.

3. You have posted this question twice. That is strictly against forum rules.
But I will respond this way. There is only one way that the implication is problematic.
That is to prove that $\left( {S \cup T} \right)^\prime \subseteq S' \cup T'$.
If $x$ is limit point of $S\cup T$ and $x$ is limit point of $S$ then we are done.

But if and $x$ is not limit point of $S$ you must prove that $x$ must a limit point of $t$.