Let $\overline{A},\overline{B}$ denote the closure of a set $A,B\subseteq \mathbb{R}$ respectively.

Prove or disprove that \begin{align*} \bigcup_{n=1}^{\infty}\overline{A_{n}} = \overline{\bigcup_{n=1}^{\infty} A_n}\end{align*}

Firstly, i can prove that $\overline{A\cup B}=\overline{A}\cup \overline{B}$.

I then can use this fact to prove that the main statement is true by Mathematical Induction.

However, i can also find a counter-example by letting $A_{n}=[\frac{1}{n},1].$

So, i think the method by Mathematical Induction must be wrong. But i could not see how so.