confirmation on direct/inverse limits

Question

Denote by $\lim$ both $\varprojlim$ and $\varinjlim$. Say that for every $n$ the $G_n$'s are groups. Assume that $\lim_{n\ge m} G_n\simeq G$ for some $m$. Is it true that also $\lim_nG_n\simeq G$? (i.e. we can discard the first terms). Thinking to the trivial example where the maps are given by inclusions and we have topological spaces instead of groups, then we know that the direct limit is the union while the inverse limit is the intersection, so that the claim is trivially true if I am not wrong. I guess that this should be true also for rings, groups etc. Can anyone confirm this?

Answer 1

let $ I $ be the quasi-order set.

1.If $I$ is directed(that is,$\forall i,j\in I,\exists k\in I$ such that $i,j≤k$.)$C\subset I$ is called cofinal subset of $I$ if $\forall i\in I$,$\exists j\in C$ such that $j≥i$ in $I$.If $C\subset I$ is cofinal,then $lim_{\rightarrow_{i\in I}}G_i=lim_{\rightarrow_{j\in C}}G_j$. (If $I$ is not directed,this maybe false.)

2.for any quasi-order set $I$,if $C\subset I$ is cofinal,then the $lim_{\leftarrow_{i\in I}}G_i=lim_{\leftarrow_{j\in C}}G_j$.this do not need directed set.