I am struggling to figure out how to approach proofs with LCM and GCF. I have an example problem:

prove that LCM(LCM(a, b), c) = LCM(a, b, c) = LCM(a, LCM(b, c)) for all natural numbers a, b, c.

I have started by proving that a, b, and c must divide the least common multiple and thus can each be multiplied by a unique integer to get the LCM. I also have toyed with the fact that LCM is the lowest value of the set of all common multiples of a, b, and c. Thus, LCM must be lower than any given common multiple of a, b, c. However, this is as far as I could really take it. What should be my next step(s)?