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1 PLAYING WITH THE NATURAL NUMBERS

Let $\mathbb{N}$ be the set of all natural numbers, $1,2,3,4,\dots$ . 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.

Definition 1.

Let $m, n$ be natural numbers. Their $10$ -adic distance is defined the following way: $\mbox{dist}_{(10)}(m,n)=10^{-k}$ if the last $k$ -digits of $m$ and $n$ are the same, but their $k+1$ -th digits are different. E.g. $\mbox{dist}_{(10)}(3,5)=1$ , $\mbox{dist}_{(10)}(245,1145)=\frac{1}{100}$ .

Problem 1.

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Show that $\mbox{dist}_{(10)}(m,n)=10^{-k}$ if and only if $10^{k}\mid m-n$ , but $10^{k+1}\nmid m-n$ .
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Show that $\mbox{dist}_{(10)}$ is “really” a distance, that is for $a,b,c\in\mathbb{N}$ :
$\mbox{dist}_{(10)}(a,b)+\mbox{dist}_{(10)}(b,c)\geq\mbox{dist}_{(10)}(a,c)$ triangle inequality holds.
Also, $\mbox{dist}_{(10)}(a,b)=\mbox{dist}_{(10)}(b,a)$ .
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Is it possible that for $a, b, c$ , all the distances $\mbox{dist}_{(10)}(a,b),\mbox{dist}_{(10)}(a,c),\mbox{dist}_{(10)}(b,c)$ are different?

Definition 2.

Let $\{a_{n}\}^{\infty}_{n=1}\subset\mathbb{N}$ be a sequence of natural numbers. We say that $\{a_{n}\}^{\infty}_{n=1}$ is convergent to $a\in\mathbb{N}$ in the $10$ -adic distance, if for any $\varepsilon>0$ there exists an $N>0$ such that if $n\geq N$ , then $\mbox{dist}_{(10)}(a_{n},a)\leq\varepsilon$ . We denote that by $a_{n}\to_{d}a$ . (Do not forget the little $d$ !!!)

Problem 2.

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Explain convergence using divisibility by the powers of $10$ .
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Construct a sequence $\{a_{n}\}^{\infty}_{n=1}\subset\mathbb{N}$ of different numbers such that $a_{n}\to_{d}173$ .
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Construct a sequence $\{b_{n}\}^{\infty}_{n=1}\subset\mathbb{N}$ such that $a_{n}+1\to_{d}0$ .

Problem 3. Let $\{a_{n}\}^{\infty}_{n=1}\subset\mathbb{N}$ converge to $a$ and $\{a_{n}\}^{\infty}_{n=1}\subset\mathbb{N}$ converge to $b$ (in our new sense!!). Show that $\{a_{n}+b_{n}\}^{\infty}_{n=1}$ converges to $a+b$ and $\{a_{n}b_{n}\}^{\infty}_{n=1}$ converges to $a b$ . Problem 4. Similarly to the real numbers we can define Cauchy-sequences (in the $10$ -adic sense):

Definition 3.

$\{a_{n}\}^{\infty}_{n=1}\subset\mathbb{N}$ is a Cauchy-sequence, if for any $\varepsilon>0$ there exists some $N>0$ such that if $n,m\geq N$ , then $\mbox{dist}_{(10)}(m,n)\leq\varepsilon$ .

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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?
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Show that if $\{a_{n}\}^{\infty}_{n=1}\subset\mathbb{N}$ , $\{b_{n}\}^{\infty}_{n=1}\subset\mathbb{N}$ are Cauchy-sequences then both $\{a_{n}+b_{n}\}^{\infty}_{n=1}$ and $\{a_{n}b_{n}\}^{\infty}_{n=1}$ are Cauchy-sequences.