Two isomorphisms of (algebraic) inverse limits I am having trouble seeing why the following two isomorphisms should hold for a Noetherian ring $A$ and ideals $I$,$J$ of $A$:


*

*$$\varprojlim A/(I+J)^n \cong \varprojlim A/I^n+J^n$$

*$$\varprojlim_m (\varprojlim_n A/(I^n + I^m)) \cong \varprojlim A/(I^n + J^n)$$


Any help or comments will be greatly appreciated.
 A: The isomorphism (1) comes from the fact that the systems $((I+J)^n)_n$ and $(I^n+J^n)_n$ define the same topology on $A$: 
$$ I^n+J^n \subseteq (I+J)^n, \quad (I+J)^{2n} \subseteq I^n+J^n.$$ 
Isomorphism (2) is more complicated (at least I don't have a simple proof). Fix an $m$. For any $n\ge m$, there is a canonical exact sequence
$$ 0\to J^m/(J^m\cap (I^n+J^n)) \to A/(I^n+J^n)\to A/(I^n+J^m)\to 0.$$ 
Passing to the limit, we get a canonical exact sequence
$$ 0 \to \varprojlim_n J^m/(J^m\cap (I^n+J^n)) \to \hat{A}:=\varprojlim_n A/(I^n+J^n)\to A_m:=\varprojlim_n A/(I^n+J^m)\to 0.$$ 
The exactness on the right comes from the surjectivity of $$J^m/(J^m\cap (I^{n+1}+J^{n+1}))\to J^m/(J^m\cap (I^n+J^n)).$$
As above, the $J^m\cap (I^n+J^n)$ define the same topology as the $J^m\cap (I+J)^n$. By Artin-Rees lemma, the latter define the same topology as the  $J^m(I+J)^n$. So the lefthand side term in the above exact sequence is $(J^m)^{\hat{}}=J^m \hat{A}$. Therefore $A_m\simeq \hat{A}/J^m \hat{A}$. Passing to the limit we get
$$\varprojlim_m A_m\simeq \hat{A}.$$ 
