On a theorem of Hurewicz and $K$-theory of complete discrete valuation rings

It is proved that for a complete discrete valuation ring $\mathfrak D$ of zero characteristic with residue field $k$ of positive characteristic $p$ and maximal ideal $\mathfrak M$, the natural homomorphism of $K$-groups with coefficients
$$ K_i(\mathfrak D;\mathbf Z/p^n\mathbf Z)\to\varprojlim_iK_i(\mathfrak D/\mathfrak M^j;\mathbf Z/p^n\mathbf Z) $$
is an isomorphism for all positive $i$ and $n$.
For the ring of integers $\mathfrak D$ in a local field $K/\mathbf Q_p$, the groups $K_i(\mathfrak D;\mathbf Z/p^n\mathbf Z)$ are finite.
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