- #26

- 63

- 13

## ~~~~~~~~~~~~~~~~~~ 3 ~ \to 10 \times 0.3 ##

## ~~~~~~~~~~~~~~~~~~ 21 ~ \to 10 \times 0.21 \times \frac 1 {10} ##

## ~~~~~~~~~~~~~~~~~~ 529 ~ \to 10 \times 0.529 \times \frac 1 {10^2} ##

## ~~~~~~~~~~~~~~~~~~ 4791 ~ \to 10 \times 0.4791 \times \frac 1 {10^3} ##

and so on.

This assignment assigns to all natural numbers ##k## a real number which is greater than or equal to ##\dfrac{\varepsilon_k}{k}##.

However, if in each line

**we add all the real numbers that are possible on the right**, we get the amounts smaller than

## ~~~~~~~~~~~~~~~~~~ 10 \times 9 ##

## ~~~~~~~~~~~~~~~~~~ 10 \times 9^2 \times \frac 1 {10}##

## ~~~~~~~~~~~~~~~~~~ 10 \times 9^3 \times \frac 1 {10^2}##

## ~~~~~~~~~~~~~~~~~~ 10 \times 9^4 \times \frac 1 {10^3}##

and so on.

However, if they are added, we get

## ~~~~~~~~~~~~~~~~~~ 10 \times 9 \times (1 + \frac 9 {10} + \frac {9^2} {10^2} + \frac {9^3} {10^3}+ \dots)##

what is finite.