- #1
Demon117
- 165
- 1
1. For integers m and n, let d(m,n)=0 if m=n and d(m,n) = 1/5^k otherwise, where k is the highest power of 5 that divides m-n. Show that d is indeed a metric.
2. The attempt at a solution
Here is what I have come up with:
PROOF: Clearly by definition d(m,n) = 0 iff m=n and d(m,n)>0 for all k in Z. Next, the same highest power k of 5 that divides m-n will also divide n-m. Therefore we have d(n,m) = 1/5^k, thus d(m,n) = d(m,n) when n is not equal to m. So d is symmetric and positive definite. I must finally show that the triangle inequality holds.
I am unsure how to proceed in showing that the triangle inequality holds. Can anyone help me?
2. The attempt at a solution
Here is what I have come up with:
PROOF: Clearly by definition d(m,n) = 0 iff m=n and d(m,n)>0 for all k in Z. Next, the same highest power k of 5 that divides m-n will also divide n-m. Therefore we have d(n,m) = 1/5^k, thus d(m,n) = d(m,n) when n is not equal to m. So d is symmetric and positive definite. I must finally show that the triangle inequality holds.
I am unsure how to proceed in showing that the triangle inequality holds. Can anyone help me?