Let $(R,\mathfrak{m})$ be a complete local ring, $a_{\lambda}$ be a decreasing net of ideals in $R$, indexed by a directed set. Consider the completion under $a_{\lambda}$topology $A=\underleftarrow{\lim} R/\mathfrak{a}_{\lambda}$. Is $A$ still complete under the $\mathfrak{m}$topology?
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I assume that $(R, \frak{m} )$ is a complete Noetherian local ring. Set $\frak{a} = \bigcap_\lambda \frak{a}_\lambda$. By passing to $R/\frak{a}$ we may assume that the $\frak{a}_\lambda$topology is separated. Now, we use a Theorem of Chevalley (1946) which says that in the complete Noetherian local ring the $\frak{m}$adic topology is weaker than every separated topology. Then every Cauchy sequence in the $\frak{a}_\lambda$topology is also a Cauchy sequence in the $\frak{m}$adic topology. Hence $R = \underleftarrow{\lim} R/\frak{a}_\lambda$ is complete. 

