Problem 0. This is not really a problem only a mental exercise. Do you remember the definition of convergence and Cauchy-ness from the last problem solving class? Can you see that analogy with the rationals? Let us discuss it for five-six minutes. OK, in this new world the natural numbers are playing the role of the rationals. But, who will play the role of the real numbers?
A -adic integer is an infinite (from the left !!!) sequence of digits
We view all the natural numbers as -adic integers by filling them with zeros on the left, so instead of we write . E.g. , when for all , is a -adic integer. We will denote the set of all -adic integers by .
Let . Their distance is defined the following way: if the last -digits of and are the same, but their -th digits are different (so, for integers, we have the same distances as before).
Problem 1. Show that satisfies the triangle inequality.
Let be a sequence of -adic numbers.
We say that the sequence is converging to if for any , there exists an such that if then . That is, .
We say that the sequence is Cauchy, if for any , there exists an such that if then .
Problem 2.
Explain convergence without the epsilons.
Show that for any there exists a Cauchy-sequence of natural numbers converging to .
Problem 3. Show that for any Cauchy-sequence of natural numbers there exists an such that . Problem 4. [This problem is the analogue of the famous Bolzano-Weierstrass Theorem.] Let be a sequence of integers. Show that the sequence has a convergent subsequence .