# 0.999...=1/Proof 5

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