Prove or disprove that closure of union of infinitely many sets is equivalent to union of infinitely many closure. 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. 

Induction is a little trickier than that, I actually made the same mistake when I first learned it.
Induction says that a statement is true for all $n\in \Bbb N$, but is says nothing about an infinite number. For example $A\cap B$ is open for open $A$ and $B$, so by induction any finite intersection of open sets is open. Same thing with your theorem, you have shown any closure of a finite union is the union of closures.
