Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows:
$M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and $g_{ji}=f_{ji}$ when $j≥i≥α$, $g_{ji}=0$ when $j≥i, j<α$.
I want to prove that $(M_i,g_{ji})_{i\in∆}$ is an inverse system (which seems easy) and find $\varprojlim M_i$. Can anybody be so kind as to answer this question? Thanks!