Question related to a bounded operator in a Hilbert space.

Question

The statement is "Let $A_n$ be a sequence in $B(H)$ and $A\in B(H)$ such that $\|A_n -A\|$ $\rightarrow 0$ and $n \rightarrow 0$ if $A_n$ is self adjoint then $A$ is also self-adjoint."

I understood the statement but I need one example for this.

Answer 1

Hint : $A\to A^{\star}$ is a continuous map. (Prove)

$A_n\to A\implies A_n^{\star}\to A^{\star}$

$A_n^{\star}=A_n$ and by uniqueness of limt $A^{\star}=A$