Countable annihilation sequence in an irreducible Banach module ($\pi (a_m\cdots a_1){\xi }_n=0(m>n)$)

For countably many linearly independent vectors in an infinite-dimensional irreducible complex Banach module, there are elements $a_n$ satisfying the displayed annihilation conditions while the tail images $\{\pi (a_n\cdots a_1){\xi }_j:j\ge n\}$ remain linearly independent. Use the quotient norm on an irreducible Banach module to make each finite annihilator a closed Banach subspace. The Jacobson density theorem makes each condition of independence of finitely many tail images open and dense there; the Baire category theorem intersects these countably many conditions. This is stronger than prescribing only finitely many images once.