# 0.999...=1

## Theorem

$0.999 \ldots = 1$

## Proof using Geometric Series

$0.999 \ldots = \dfrac a {1 - r}$

where $a = \dfrac 9 {10}$ and $r = \dfrac 1 {10}$.

Since our ratio is less than $1$, then we know that $\ds \sum_{n \mathop = 0}^\infty \frac 9 {10} \paren {\frac 1 {10} }^n$ must converge to:

$\dfrac a {1 - r} = \dfrac {\frac 9 {10} } {1 - \frac 1 {10} } = \dfrac {\frac 9 {10} } {\frac 9 {10} } = 1$

$\blacksquare$

## Proof using Fractions

 $\ds 0.333 \ldots$ $=$ $\ds 1 / 3$ $\ds \leadsto \ \$ $\ds 3 \paren {0.333 \ldots}$ $=$ $\ds 3 \paren {1 / 3}$ $\ds \leadsto \ \$ $\ds 0.999 \ldots$ $=$ $\ds 3 / 3$ $\ds$ $=$ $\ds 1$

$\blacksquare$

## Proof using Multiplication by 10

Let $c = 0.999 \ldots$

Then:

 $\ds c$ $=$ $\ds 0.999 \ldots$ $\ds \leadsto \ \$ $\ds 10 c$ $=$ $\ds \paren {9.999 \ldots}$ multiplying $c$ by $10$ $\ds \leadsto \ \$ $\ds 10 c - c$ $=$ $\ds \paren {9.999 \ldots} - \paren {0.999 \ldots}$ subtracting $c$ from each side $\ds \leadsto \ \$ $\ds 9 c$ $=$ $\ds 9$ $\ds \leadsto \ \$ $\ds c$ $=$ $\ds 1$

It follows that:

$0.999 \ldots = 1$

$\blacksquare$

## Proof using Long Division

We begin with the knowledge that:

 $\ds \frac 9 9$ $=$ $\ds \frac 1 1 = 1$

Now we divide $9$ by $9$ using the standard process of long division, only instead of stating that $90$ divided by $9$ is $10$, we say that it is "$9$ remainder $9$," yielding the following result:

$\require{enclose} \phantom{0..}0.9999\ldots\\ 9\enclose{longdiv}{9.0000\ldots}\\ \phantom{0}\underline{\phantom{0}8.1}\\ \phantom{000.}90\\ \phantom{0.}\underline{\phantom{00}81}\\ \phantom{0000.}90\\ \phantom{0.}\underline{\phantom{000}81}\\ \phantom{00000.}9\ldots\\$

Thus, we are compelled to believe that:

$0.999\ldots = \dfrac 9 9 = 1$

$\blacksquare$

## Proof using Sequences

 $\text {(1)}: \quad$ $\ds 0 . \underset n {\underbrace {999 \cdots 9} }$ $=$ $\ds 1 - 0.1^n$ $\ds 0.999 \cdots$ $=$ $\ds \eqclass {\sequence {0.9, \, 0.99, \, 0.999, \, \cdots} } {}$ Definition of Real Numbers $\ds$ $=$ $\ds \eqclass {\sequence {1 - 0.1^1, \, 1 - 0.1^2, \, 1 - 0.1^3, \, \cdots} } {}$ from $(1)$ $\ds$ $=$ $\ds \eqclass {\sequence {1, \, 1, \, 1, \, \cdots} } {} - \eqclass {\sequence {0.1^1, \, 0.1^2, \, 0.1^3, \, \cdots} } {}$ $\ds$ $=$ $\ds 1 - 0$ Sequence of Powers of Number less than One, Definition of Real Numbers $\ds$ $=$ $\ds 1$

$\blacksquare$