0.999...

There is a common argument as to whether 0.9999~ equals 1 or not. It does. Convergence is the primary reason as to why.

$0.9999~$

$9*0.1111~$

$9*(1/10+1/100+1/1000+...)$

$9*(-1+1+1/10+1/100+1/1000+...)$

$9*(-1+(1/(1-1/10)))$

$9*(-1+(10/9))$

$-9+10$

$1$