Let $M_\lambda $, $\lambda \in \Lambda $ be a fundamental system of open submodules. Then $N \cap M_\lambda $ is a fundamental system of open submodules of $N$ and $M_\lambda + N/N$ is a fundamental system of open submodules of $M/N$. Thus we see that (1) follows from the exactness of the sequences
\[ 0 \to N/N \cap M_\lambda \to M/M_\lambda \to M/(M_\lambda + N) \to 0 \]
and the fact that taking limits commutes with limits. The second statement follows from this and the fact that $N \to N^\wedge $ has dense image and that the kernel of $M^\wedge \to (M/N)^\wedge $ is closed.