Finding a counter example for $ \left(A+A\right)'\subseteq\left(A'+A\right)\cup\left(A'+A'\right)$

Question

let $A'$ be the set of limit points of $A\subseteq\mathbb{R}$ and $A+B=\{x+y:x\in A,y\in B\}$, I'm required to find a counter-example for:

$$ \left(A+A\right)'\subseteq\left(A'+A\right)\cup\left(A'+A'\right)$$

But no matter how hard I try I can't seem to find any, I tried proving this is true instead, to maybe find where my logic needs to focus, but I wasn't able to make any progress that way.

The problem is that for any set I find, I can't "generate" a new limit point that is not the sum of two limit points, or the sum of an element and a limit point.

i.e. $\{\frac{1}{n}\}'=\{0\}$, therefore $\left(\{\frac{1}{n}\}+\{\frac{1}{n}\}\right)'=\{0,\frac{1}{n}\}$, but $0\in \{0\} + \{0\}$ and $\forall n:\frac{1}{n}\in\{\frac{1}{n}\}+\{0\}'$

Any tips on finding a counter example?

Answer 1

Note that the right hand side is empty if $A'=\emptyset$. So it is tempting to try and find an $A$ without limit points such that $A+A$ does have limit points. The set $A$ must be closed and discrete. This could be achieved by picking one point from every interval $[n,n+1]$ for integer $n$. Can you see a way of doing this such that the sum $A+A$ has $0$ as a limit point? Hint: In $(-\infty,0]$, this set is just $-\Bbb N$.

You can hover over the shaded area to reveal the solution.

$$A=\left\{n+\frac1{\max(n,0)+1}\middle|\ n\in\Bbb Z\right\}$$