Natural numbers distributive lattice

[Xael]

I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with gcd as AND and lcm as OR.

I know it can be shown that a AND (b OR c) >= (a AND b) OR (a AND c) for a general lattice, and that if we can show the opposite, that a AND (b OR c) <= (a AND b) OR (a AND c) that implies the two are equal. How do I prove this second part? I am not experienced with number theory, and I have struggled to get a meaningful expression of gcd's and lcm's.

Alternatively, is there a different way you can show me how to prove this?

Thank you!

 

[nomadreid]

The answer appears in http://math.stackexchange.com/quest...red-by-divisibility-is-a-distributive-lattice, whereby your question there, as to whether there is a bound on the value of the entries of the sequence: every sequence is bounded, since remember the condition that all but a finite number of the sequences referred to there are 0.