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We give some reasonable and usable conditions on a sequence of norm one in adual banach space under which the sequence does not converges to the origin inthe $w^*$-topology. These requirements help to ensure that the Lagrangemultipliers are nontrivial, when we are interested for example on the infinitedimensional infinite-horizon Pontryagin Principles for discrete-time problems.

A usufel lemma for Lagrange multiplier rules in infinite dimension