Let be the set of all natural numbers, . We define a notion of “distance” on the natural numbers based on their similarities. Two natural numbers are “close” if they are very similar to each other looking at them from the right.
Let be natural numbers. Their -adic distance is defined the following way: if the last -digits of and are the same, but their -th digits are different. E.g. , .
Problem 1.
Show that if and only if , but .
Show that is “really” a distance, that is for :
triangle inequality holds.
Also, .
Is it possible that for , all the distances are different?
Let be a sequence of natural numbers. We say that is convergent to in the -adic distance, if for any there exists an such that if , then . We denote that by . (Do not forget the little !!!)
Problem 2.
Explain convergence using divisibility by the powers of .
Construct a sequence of different numbers such that .
Construct a sequence such that .
Problem 3. Let converge to and converge to (in our new sense!!). Show that converges to and converges to . Problem 4. Similarly to the real numbers we can define Cauchy-sequences (in the -adic sense):
is a Cauchy-sequence, if for any there exists some such that if , then .
How can you visualize “Cauchy-ness”? Can you explain it without using epsilons? Could you give some examples of Cauchy-sequences of natural numbers that are not convergent?
Show that if , are Cauchy-sequences then both and are Cauchy-sequences.