Let $u\neq 0$ be any element of Boolean Algebra. Let $u_1,u_2,u_3,\dots,u_m$ be all atoms in $A$ such that $u_1\leq u,u_2 \leq u,\dots, u_m\leq u$.
Prove that $u_1\vee u_2\vee u_3\vee\dots\vee u_m = u$.
Here '$\vee$' denotes join of a lattice. I tried to prove the equality by proving LHS $\leq$ RHS AND RHS $\leq$ LHS, so LHS $=$ RHS. Proving from right to left was simple. Since $u_1\leq u, u_2\leq u,\dots,u_m\leq u$, we can say that $$u_1+u_2+u_3+\dots+u_m\leq \underbrace{u+u+\dots+u}_{m\text{ times}} = u$$ (since closed under Boolean algebra). But I am not able to prove from left to right.
Please help me with a complete solution if possible, since I am beginner in this course.