I asked this question to the math.stackexchange but couldn't get an answer.

Let $A$ be a closed convex process from $R^n$ to $R^n$, $I$ be the identity map, $\lambda$ be a real number, and $k$ be a positive integer. It is obvious that $A - \lambda I$ and $(A - \lambda I)^{-1}$ are closed but I cannot find a way to prove that $(A - \lambda I)^{-k}$ is closed. This result is stated or used without a proof in [this book][1] (Theorem 2.14) and [this book][2] (pp. 176).

If we can assume that the domain or the range of $A$ is bounded, the proof is staightforward but not in other case. Then, I tried to use a condition found from Theorem 39.8 of Rockafellar's Convex Analysis:
> Let $A$ and $B$ be convex processes from $R^n$ to $R^m$ and from $R^m$ to $R^p$, respectively. If $A$ and $B$ are closed and $\mathrm{ri}(\mathrm{range}B^*)$ meets $\mathrm{ri}(\mathrm{dom}A^*)$, then $BA$ is closed.

but couldn't make any progress. Does anybody know where I can find the proof of it?


  [1]: https://books.google.de/books?id=6EUPCgAAQBAJ&pg=PA213&lpg=PA213&dq=introduction+to+the+theory+of+differential+inclusions&source=bl&ots=9avPygrjjG&sig=VI1V-YSS5GY6IFRlinIK7G8zfHE&hl=en&sa=X&ved=0ahUKEwjAkdaT04TQAhXHwBQKHRR9BmoQ6AEILDAD#v=onepage&q=introduction%20to%20the%20theory%20of%20differential%20inclusions&f=false
  [2]: https://books.google.de/books?id=CPh6DAAAQBAJ&pg=PA168&lpg=PA168&dq=adjoint%20of%20convex%20processes&source=bl&ots=9jXh_TM-CP&sig=OK0-ENA87CsylGz8xEmFPbGShUo&hl=en&sa=X&ved=0ahUKEwjin4Dy0oTQAhVB6RQKHRMHCREQ6AEITDAJ#v=onepage&q=adjoint%20of%20convex%20processes&f=false