Under the assumption that the invariant subspace problem has a positive solution, this is an immediate consequence of Lemma 7.1.11 (The Triangularization Lemma) in Radjavi and Rosenthal's book Simultaneous Triangularization:

If $P$ is a property of families of operators that is inherited by quotients, and if every family satisfying $P$ (and acting on a space of dimension greater than 1) has a non-trivial closed invariant subspace, then every family satisfying $P$ is triangularizable in the sense that there is a maximal chain of closed subspaces, each of which is invariant for the family.

Under the assumption that the invariant subspace problem has a positive solution, this is an immediate consequence of Lemma 7.1.11 (The Triangularization Lemma) in Radjavi and Rosenthal's book Simultaneous Triangularization:

Say that a property $P$ of families of operators is inherited by quotients if whenever $N \subseteq M$ are invariant subspaces for a family of operators with $P$, then the family of induced operators on the quotient $M/N$ also has $P$.

Then if $P$ is a property of families of operators that is inherited by quotients, and if every family of operators satisfying $P$ (and acting on a space of dimension greater than 1) has a non-trivial closed invariant subspace, then every family satisfying $P$ is triangularizable in the sense that there is a maximal chain of closed subspaces, each of which is invariant for the family.

Under the assumption that the invariant subspace problem has a positive solution, this is an immediate consequence of Lemma 7.1.11 (The Triangularization Lemma) in Radjavi and Rosenthal's book Simultaneous Triangularization:

If P is a property of families of operators that is inherited by quotients, and if every family satisfying P (and acting on a space of dimension greater than 1) has a non-trivial closed invariant subspace, then every family satisfying P is triangularizable in the sense that there is a maximal chain of closed subspaces, each of which is invariant for the family.

Under the assumption that the invariant subspace problem has a positive solution, this is an immediate consequence of Lemma 7.1.11 (The Triangularization Lemma) in Radjavi and Rosenthal's book Simultaneous Triangularization:

If $P$ is a property of families of operators that is inherited by quotients, and if every family satisfying $P$ (and acting on a space of dimension greater than 1) has a non-trivial closed invariant subspace, then every family satisfying $P$ is triangularizable in the sense that there is a maximal chain of closed subspaces, each of which is invariant for the family.