Division Algorithm - For every pair of natural numbers m,n in N with n!=0, there is a unique q,r in N, such that m=qn+r and r<n.

Consider the Set A of natural numbers that might be suitable candidates for r, in other words, A={r' in N: r' = m - nq' >=0 for some q' in N}. Show that A must be non empty and use the Well-ordering principle to show that there in an element r in A that satisfies r<n.

I am trying to show that A must be non empty and use the Well-ordering principle to show that there in an element r in A that satisfies r<n.

This is my incomplete informal attempt:

1.

m,n in N

m/n <= n [since n!=0]

2.

q,r in N

m=qn+r

(m-r)/q=n

m/n <= (m-r)/q [combining 1 and 2]

qm/n<=m-r

r<=m-(qm)/n

0<=r<=m-(qm)/n [since r in N]

Does this show that there is at least 1 element in A?