I'm having some trouble finding the set of all accumulation points of $A=[0,1]$. I already prove that $0$ and $1$ are accumulation points, but I can't find a way to prove that any $p \in(0,1)$ is also an accumulation point.
If anyone could help me, it would be great.
Thank you very much!
Best Answer
Every point of $[0,1]$ is an accumulation point. To see this, let $x\in[0,1]$. If $\varepsilon>0$, do you understand why the intersection $\big((x-\varepsilon,x+\varepsilon)\setminus\{x\}\big)\cap[0,1]$ is non-empty? If so, you understand that $x$ is an accumulation point of $[0,1]$.
On the other hand, any point outside $[0,1]$ is not an accumulation point. Indeed, if $x\not\in[0,1]$, then we can find $\delta>0$ so that $(x-\delta,x+\delta)\cap[0,1]=\emptyset$.