Flaw in the proof that $(-1,1)$ is closed in $\mathbb{R}$ This is an answer given by a student. I am totally lost with this solution, could not convince him that this does not work.
He said write $\mathbb{R}$ as $$\bigcup_{r\in \mathbb{R} }r (-1,1).$$
Each $r(-1,1)$ is open in $\mathbb{R}$ and so is the union over all real numbers except when $r=1$. 
So, $$(-1,1)=\mathbb{R}\setminus \bigcup_{r\neq 1} r(-1,1)$$
which implies that $(-1,1)$ is closed in $\mathbb{R}$ being complement of an open set.
How do I convince him that this is wrong justification. I was shocked when I saw this for the first time and still I shock. Could not spend more time on this, so, posting it here.

The equlity from your post is false. Actually$$\mathbb{R}\setminus\bigcup_{r\neq1}r(-1,1)=\emptyset$$(which is indeed a closed set).
