Why $0.999$... isn't the largest number before 1?

Question

Why doesn't it called like that? It seems fair, $1$ called $1$ while $0.999$... being the largest number before $1$, and not called $1$ while not look like it is. Let's say it isn't, how would that number look like?

Answer 1

$0.999999\ldots$ is exactly $1$. It is just another way of writing the number $1$, and is not less than $1$. You can prove this by subtracting $ 1 - 0.999999 \ldots $ and seeing that the answer is $0$. Alternatively, you can prove that there is no "largest number less than 1."

Proof. Note that for any fraction $\frac{p}{q}$ less than one, there is a slightly bigger fraction which is still less than one; in particular, $\frac{p+q}{2q}$ is a fraction such that $\frac{p}{q} < \frac{p+q}{2q} < 1$. $\Box$