I need an help with the following exercise about Yosida approximations. Let's fix the notation: let $A$ be an unbounded linear operator define on a dense domain $D(A)\subset X$ of a Banach space $X$.
Define the resolvents $J_n:=(I-\frac{1}{n}A)^{-1}$ and the Yosida approximations $A_n:=n(J_n-I)$.
We have already proved that, if $x\in D(A)$, $||A_n x||\to ||A x||$ as $n\to \infty.$ Our aim is to prove that, if $x\notin D(A)$ then the sequence $\{||A_nx||\}$ is increasing and unbounded, i.e. $||A_nx||\to \infty$.
How can we prove it? We have tried to prove it directly by definition of $A_n$, but we got stuck. Any help is really appreciated.