Suppose $A$ is an open set in $\mathbb{R}$. Show that $\mathbb{I}_A$ is a lower semi continuous function where, a function, $f:\mathbb{R}\rightarrow\mathbb{R}$ is said to be lower semi continuous at $x$ if $\lim\inf_{y\rightarrow x}f(y)\ge f(x)$.
Now, what I need to prove is that for any $x\in\mathbb{R} $, $\lim\inf_{y\rightarrow x}\mathbb{I}_A(y)\ge \mathbb{I}_A(x)$. If I can show that the LHS is necessarily $1$ for $x\in A$, then I should be done. In that case, RHS is $1$. Now, further dividing this into $x\in A^0$ and $x\in \partial A$, I am done with all interior points of $A$. I'm not sure what to do with the remaining part. Is this proof even correct?