[math]x = 0.99\bar{9}[/math]
[math]10x = 9.99\bar{9}[/math]
[math]10x - x = 9.99\bar{9} - 0.99\bar{9}[/math]
[math]9x = 9[/math]
[math]x=1[/math]
but earlier we said that
[math]x = 0.99\bar{9}[/math]
therefore,
[math]0.99\bar{9}=1[/math]
end of proof