My attempt: Let $\{v_1, v_2,...,v_m\}$ is the set of atoms in a finite lattice L. Now the chain
$0<v_1 \lt (v_1 \lor v_2) \lt (v_1 \lor v_2 \lor v_3) \lt...\lt (v_1\lor v_2 \lor ...\lor v_m) \leq I$
is in L. And the length of this chain is less than or equal to m.