TITLE: Prove that $\lfloor0.999\dots\rfloor= ?$ $0$ or $1$?
QUESTION [21 upvotes]: I think $\lfloor0.999\dots\rfloor= 1$, as $0.999\dots=1$,but I have doubt, as $\lfloor0.9\rfloor=0$,$\lfloor0.99\rfloor=0$,$\lfloor0.9999999\rfloor=0$, etc.

REPLY [11 votes]: For one thing, $0.999...$ is exactly equal to $1$.
To prove that $0.999... = 1$, we use sums of infinite geometric sequences. We know that
$$\sum_{k=0}^\infty r^k = \frac{1}{1-r} \forall \;\left|r\right|\lt1$$
It is fairly simple to prove this statment, although I won't go into that.
For our specific instance, we have
$$0.999... = \frac{9}{10}\sum_{k=0}^\infty (\frac{1}{10})^k = \frac{9}{10} \cdot\frac{1}{1-\frac{1}{10}} = \frac{9}{10}\cdot\frac{10}{9}=1$$
Using the substitution property, we can substitute any alternate representation of a number into an equation, and it will yield the same result. Therefore,
$$\lfloor0.999...\rfloor = \lfloor1\rfloor = 1$$