Join Irreducible elements in Finite Lattice.

Question

If $ L $ is a finite lattice then how can we show that each element is of the form $ a_1\vee a_2\vee a_3...\vee a_k$ where each $ a_i$ is a join-irreducible element.

Answer 1

By contradiction. Suppose that some element isn't of that form. Then you can construct an infinite descending chain.

Start with an element which is not the join of the join-irreducible elements beneath it. Then it itself cannot be join-irreducible, so you can express it as the join of two other elements.

You should be able to take it from there.