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2 Infinite numbers a.k.a the 10-adics

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?

Definition 4.

A 10-adic integer x¯ is an infinite (from the left !!!) sequence of digits

x¯=…⁢x5⁢x4⁢x3⁢x2⁢x1

We view all the natural numbers as 10-adic integers by filling them with zeros on the left, so instead of 452 we write …⁢00000000000000000000452. E.g. x¯=…⁢1111111, when for all k≥1, xk=1 is a 10-adic integer. We will denote the set of all 10-adic integers by ℤ(10).

Definition 5.

Let x¯,y¯∈Z(10). Their distance 𝑑𝑖𝑠𝑡(10)⁢(x¯,y¯) is defined the following way: 𝑑𝑖𝑠𝑡(10)⁢(x¯,y¯)=10-k if the last k-digits of x¯ and y¯ are the same, but their k+1-th digits are different (so, for integers, we have the same distances as before).

Problem 1. Show that dist(10) satisfies the triangle inequality.

Definition 6.

Let {x¯n}n=1∞⊂Z(10) be a sequence of 10-adic numbers.

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    We say that the sequence {x¯n}n=1∞ is converging to y¯∈ℤ(10) if for any ε>0, there exists an N>0 such that if n≥N then 𝑑𝑖𝑠𝑡(10)⁢(x¯n,y¯)<ε. That is, x¯n→dy¯.

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    We say that the sequence {x¯n}n=1∞ is Cauchy, if for any ε>0, there exists an N>0 such that if m,n≥N then 𝑑𝑖𝑠𝑡(10)⁢(x¯n,x¯m)<ε.

Problem 2.

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    Explain convergence without the epsilons.

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    Show that for any x¯∈ℤ(10) there exists a Cauchy-sequence of natural numbers converging to x¯.

Problem 3. Show that for any Cauchy-sequence of natural numbers {an}n=1∞ there exists an x¯∈ℤ(10) such that an→dx¯. Problem 4. [This problem is the analogue of the famous Bolzano-Weierstrass Theorem.] Let {an}n=1∞ be a sequence of integers. Show that the sequence has a convergent subsequence {ank}k=1∞.