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Let R be a commutative Noetherian ring and M a finitely generated R-module. We show in this paper that, for an integer t, if the local cohomology module H_aⁱ(M) with respect to an ideal a is finitely generated for all i<t, then for all a-filter regular elements x contained in a enough large power of a and all i<t−1. As consequences we obtain generalizations, by very short proofs, of the main results of M. Brodmann and A.L. Faghani [M. Brodmann, A.L. Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000) 2851–2853] and of H.L. Truong and the first author [N.T. Cuong, H.L. Truong, Asymptotic behavior of parameter ideals in generalized Cohen–Macaulay module, J. Algebra 320 (2008) 158–168].