Answer :
The natural numbers are well-ordered: which means every set of natural numbers has a least element.
So suppose S is a set of natural numbers closed under addition.
Let k be the smallest element of S.
Then S contains:
k,k+k, k+k+k,....etc
in other words S must contain all multiples of k.
could S contain other elements besides multiples of k?
suppose it did. suppose it contained m.
then we get all natural numbers of the form ak + bm.
for example, if k = 2, m = 3, S might be:
S = {2,3,4,5,6,7,8,.......} = N - {0,1}.
note we can write this set as:
{2 + k(gcd(2,3)): k in N}
this can be generalized to more than a pair of numbers
So suppose S is a set of natural numbers closed under addition.
Let k be the smallest element of S.
Then S contains:
k,k+k, k+k+k,....etc
in other words S must contain all multiples of k.
could S contain other elements besides multiples of k?
suppose it did. suppose it contained m.
then we get all natural numbers of the form ak + bm.
for example, if k = 2, m = 3, S might be:
S = {2,3,4,5,6,7,8,.......} = N - {0,1}.
note we can write this set as:
{2 + k(gcd(2,3)): k in N}
this can be generalized to more than a pair of numbers
